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Source Share this: Tweet. Like this: Like Loading Previous TR Reel Builder. You may have missed. Fly Fishing news. Of three pipes, the first and second running together can fill a swimming pool in 80 min. How long will it take each pipe running alone to fill it? At what time are the hands of a clock together between 2 and 3 o'clock? At what time between 2 and 3 o'clock are the hands of a clock pointing in opposite directions? At what time between 6 and 7 o'clock are the hands of a clock at right angles to each other?

A man having 10 hr. Find the distance he rides. The length of a given rectangle exceeds the width by 14 in. But if the length is diminished by 9 in. Find the dimensions of the given rectangle. Find his capital. A party of boys purchased a motor boat.

How many boys were there and what did the boat cost? A lb. A mass of copper and tin weighing lb. If the specific gravity of copper is 8. The denominator of a certain fraction exceeds the numerator by 2. Find the original fraction. In a given number of two digits, the units' digit is double the tens' digit.

If the position of the digits is re- versed, the value of the fraction is increased by Find the number. In a given number of three digits, the sum of the digits is 9. The units' digit exceeds the hundreds' digit by 1. If tlie units' and hundreds' digits exchange places, the value of the number is increased by A steamer can run 20 miles an hour in still water. If the steamer can go 72 mi, with a current in the same time that it can go 48 mi.

What was his original salary per month? How many dimes and how many quarters are there? A and B together can do a certain piece of work in 10 days ; but at the end of 7 days A stops working and B finishes it by working alone for 5 days. How long would it take each man, working alone, to do the entire work? A farmer has enough feed for his oxen to last a certain number of days. If he sold 10 oxen, his feed would last 30 days longer. Find how many oxen he has and for how many days he has feed.

Given three metals of the following composition by weight: the first, 5 parts gold, 2 silver, 1 lead; the second, 2 parts gold, 6 silver, 1 lead ; the third,. To obtain 9 ounces of a metal con- taining equal quantities by weight of gold, silver, and lead, how many ounces of the first, second, and third must be melted together? The sum of two numbers is a, and five times the smaller number exceeds four times the larger by b. If A can do in p days a piece of work which B can do in q days, how long will it take them together to do the work?

Find three consecutive numbers whose sum is a. Generalize Ex. Who first used the letters a, J, c, to represent known numbers? Tell all you can about this man. Before the use of a, J,? Discuss the rela- tive advantages in these different sets of symbols. Also find w in terms of A and L 2. Also solve for t For p. Give the meaning of each formula obtained. Find each letter in terms of the others, in the following formulas used in mechanics and physics.

If possible, give the meaning of the formulas obtained : 9. What is the equivalent temperature on the Fahrenheit scale? What temperature is numerically the same on both the Centigrade and Fahrenheit scales? Find that temperature on the Fahrenheit scale which is numerically double the equivalent temperature on the Centigrade scale. Also find that temperature on the Fahrenheit scale which is numerically one half the equivalent temperature on the Centigrade scale.

How many examples in Exercise 6 p. A Variable is a quantity which has an indefinite number of different values. A function is a variable which depends on another variable for its value. Thus, the area of a circle is a function of the radius of the circle ; the wages which a laborer receives is a function of the time that the man works. A function may depend for its value on more than one variable.

Thus, the area of a rectangle depends on two quantities — the length of the rectangle and the breadth. The present treatment of graphs, however, is limited to functions which depend on a single variable. In algebra, we study only those functions which have a definite value for each definite value of the variable. Uses of Oraphs. A graph is useful in showing at a glance the place where the function represented has the greatest or least value and where it is changing its value most rapidly, and in making evident similar properties of the function.

Graphs of algebraic equations are useful in making evi- dent certain properties of equations which are otherwise difficult to understand, A graph also often furnishes a rapid method of determining the root or roots of an equation. Framework of Seference. Axes are two straight lines perpendicular to each other which are used as an auxiliary framework in constructing graphs; as XX' and FF'. The X-axis, or axis of abscissas, is the horizontal axis ; as XX'.

The ordinate of a point is the line drawn from the given point parallel to the y-axis and termi- nated by the a:-axis. The abscissa of a point s- Y' is the part of the a;- axis intercepted between the origin and the ordinate. Abscissas to the right of the origin are plus ; those to the left are minus. They are usually written together in parenthesis with the abscissa first and a comma between. Thus, the point 2, 4 is the point whose abscissa is 2 and ordi- nate 4, or the point P of the figure.

Similarly, the point — 3, 2 is Q; - 2, - 2 is -R; and 1, - 4 is 5. The quadrants are the four parts into which the axes di- vide a plane. Construct the triangle whose vertices are 3, 4 , -2,3 , -1, Construct the quadrilateral whose vertices are -3,4 , -2, -6 , 4, -3 , 1,4. All the points on the a:-axis have what ordinate?

Construct the rectangle whose vertices are —4, 4 , 8, 4 , -4, -2 , 3, -2 , and find the number of square spaces in its area. Construct the triangle whose vertices are 3, — 3 , —4, — 3 , —1, 5 and find the number of square spaces in its area. Construct the triangle whose vertices are —1, 4 , 5, — 4 , 1, — 1 , and find the length of its sides. In which quadrant are the abscissa and ordinate both positive? Both negative? In which quadrant is the abscissa negative and the ordinate positive?

In which is the abscissa positive and the ordinate negative? Graphs of Equations of the First Degree Linear Equations. It will always be found that the graph of an equation of the first degree which contains not more than two unknown quantities is a straight line. Hence, A linear equation is an equation of the first degree.

Abbreviated Method of constructing the Graph of a Linear Equation. Since a straight line is determined by- two points, in order to construct the graph of an equation of the first degree, it is sufficient to construct any two points of the graph and draw a straight line through them. Hence, the graph passes through the points 0, — 3 and 2, 0 , or CD is the required graph. The greater the dis- tance between the points chosen, the more accurate the con- struction will be.

It is usually advisable to test the result obtained by locating a third point and observing whether it falls upon the graph as constructed. If the given line does not pass through the origin, or near the origin on both axes, it is usually best to construct the line by determining the points where the line crosses the axes as above.

Hence, the graph passes close to the origin on both axes. Let the pupil construct the figure. To the axis of y? Oraphic Solution of Simnltaneoiu Linear Equations. Graphing two simultaneous equations is a conven- ient method of testing or checking their algebraic solu- tion. You wiU find that the graphs obtained are parallel straight lines.

Now try to solve the same equations algebraically. You will find that when either x ot y is eliminated, the other unknown quantity is eliminated also, and that it is therefore impossible to obtain a solution. The reason why an algebraic solution is impossible is made clear by the fact that the graphs, being parallel lines, cannot intersect ; that is to say, there are no values of x and y which will satisfy both of these lines, or both equations, at the same time.

Construct the graphs of 8. Give reasons for your answer. Also an example similar to Ex. How many examples in Exercise 9 p. Oraphio Solntion of Written Problems. Kailway Problems. Two places, A and B, are miles apart. At a given time a train leaves each of the two places and travels toward the other, the train from A at 40 miles an hour and the train from B at 20 miles.

In how many hours will they meet and how many miles from A? The train dispatcher represents the distance between the stations by the line A By each space denoting 10 miles. He obtains the distance from A at which the trains meet, by measuring AD to scale and hence determines the siding at which one train must wait for the other.

He obtains the time that elapses before the trains meet, by measuring CD to scale. The advantage of the graphical method is that in this solution it is easy to make allowance for any waits which trains may make at stations.

Hence, railroad time-tables are often constructed entirely by graphical methods. Problems in the Mixture of Materials. Graphical Solution We construct a rectangle, and write in two adjacent comers here the left-hand corners the per cents of fat 30 and 5 in the two 1 8 S 30 25 20 5 5 GRAPHS 77 given fluids ; and in the middle of the rec- tangle we write the per cent 25 desired in the mixture.

The differences between the number in the middle and the numbers in the corners 20 and 5 are then found and placed as in the diagram. The differences thus found show the rela- tive amounts of the given fluids to be used, viz. Now solve this problem algebraically by the method used in solving Ex. The distance between New York and Philadelphia is 90 miles. If a train leaves New York at noon and goes 80 miles an hour, and another train leaves Phila- delphia at the same time and goes 20 miles an hour, at what time will they meet?

How far from New York will they meet? The distance from New York to Boston by a certain route is miles. If a train leaves Boston at 2 p. The distance from A to B is 36 miles. At the same time, an- other boy starts from B on a bicycle and rides toward A at the rate of 12 miles an hour, but at the end of each hour of riding he rests J hour.

By means of a graph, determine where and when the two boys will meet. State and illustrate some of the advantages connected with the use of algebraic symbols. State and illustrate some of the advantages connected with the use of elementary algebraic processes. Give the value of 1. Divide each of the following by 2: 1? By 4a. Give the reciprocal of each of the fractions named in Ex. Involution is the operation of raising an expression to any required power. Since a power is the product of equal factors, inyolution is a species of multiplication.

In this multiplication, the fact that the quantities multiplied are equal leads to important abbreviations of the work. Powers of Monomials Law of Exponents or Index Law. This law enables us to reduce the process of finding the power of a product to the simpler process of finding the power of each factor of the given product. Law of Signs. It is evident from the law of signs in multiplication that 1 An even power of a quantity whether pluB or minm is always positive.

Involntion of Monomials in General. Hence, to raise a monomial to a required power, Haise the coefficient to the required power ; Multiply the exponent of each literal factor by the index of the required power ; Prefix the proper sign to the result. Powers of Fractions. By a method similar to that used in Art. General Method. In obtaining a required power of a binomial, it is possible to abbreviate the work even more than in the involution of a monomial.

It is sufficient, in taking up the subject here for the first time, to obtain several powers of a binomial by actual mul- tiplication, and by comparing them, to obtain a general method for writing out the power of any binomial. A formal proof of the method is given later. Comparing the results obtained, it is perceived that I. The number of terms equals the exponent of the power of the binomial, plus one. The exponent of a in the first term equals the index of the required power, and diminishes by 1 in each succeeding term.

The exponent of b in the second term is 1, and increases by 1 in each succeeding term. The coefficient of the first term is 1; the coefficient of the second term is the index of the re- quired power. From any given term, to find the coefficient of the next term, multiply the coefficient of the given term by the exponent of a in that term and divide by the number of the given term, IV. Signs of Terms. If the binomial is a difiference, the signs of the even terms are minus ; otherwise the signs of all the terms are plus.

Observe that the coefficients of the latter half of the expansion are the same as those of the Jirst half in reverse order. Binomials with Complex Terms. If the terms of the given binomial have coefficients or exponents other than unity, it is usually best to separate into two steps the process of writing out the required power. Applioatioii to Polynomials. Give the value of 3 x On squared paper show the meaning of. Also of 1.

Is there any advantage in knowing this relation? Expand and check : Alsoof a-a; i8. Give orally the value of the various powers of A Boot of a given quantity is a quantity which, taken as a factor a certain number of times, will produce the given quantity.

EvolatioiL is the process of finding a required root of a quantity. What is the radical or root sign? What is the mean- ing of V9? Homber of Boots. Taking a particular example, we find that Vi has two values, viz. A number containing a square root of a negative quan- tity is termed an imaginary number. A real number is a number which does not contain an imaginary number. The nature of the sqaare of an imaginary nnmber, as of V— 4, is explained in Chapter IX p.

If we include imaginary roots, it may be shown that when any root of a given number is extracted, the number of possible roots equals the index of the root to be extracted. Thus, in taking the cube root of 8, we find three possible roots, viz. The Principal Boot of a number is that real root of the number which has the same sign as the number itself. In this chapter, only the principal roots of the numbers are con- sidered.

Evolution op Monomials Index Law. Hence, to extract a required root of any monomial, Extract the required root of the coefficient; Divide the exponent of each letter by the index of the re- quired root; Prefix the proper sign to the result. How may the work be checked?

Write the value of 7. Find the largest cube which is a factor of each of the following : 40, , 88, , , , Extract the square root of each of the following by tak- ing out pairs of like factors or square factors : Square Root Square Boot of Polynomials.

Evidently we may reverse this process, and extract a square root to three terms by regarding two terms of the root, when found, as a single quantity. Similarly, a fourth term of a root, or any number of terms, may be obtained by regarding the root already found as a single quantity. Extract the square root of a! Let the pupil state the above process as a rule. Square Boot of Arithmetical Hiunhers. The same general method as that used in Art. The details of the method of extracting the square root of num- bers are explained in arithmetic see DureWs Advanced Arithmetic , As an illustration of the process, we give the following example : Ex.

Extract the square root of J9 For extraction of Cube Root see Appendix, p. VsTVl Find the altitude of an equilateral triangle whose side is 27 in. Find the side of an equilateral triangle whose alti- tude is 27 in. Find the side of a square whose diagonal is 36 in. If a city park is yd. How much time does a man save in a year by walking thus, if he walks at the rate of 4 mi. What is the meaning of this process in geometry?

Find in feet the radius of a circle whose area is one square rod. Find in feet the length of the tether by which a cow must be tied in order that she may graze over two fifths of an acre. What is the meaning of this process as applied to the sphere?

The area of California is , sq. Find the side of a square having an equivalent area. How may you then visualize the area of California? By the use of the method of Ex. A bushel measure is to be a square 12 inches deep, Find in inches the inside edge of the square top.

How many examples in Exercise 14 p. Fodtiye Integral Exponents. Fractional and Hegatiye Exponents. We have seen that by using fractions as well as integers, and negative as well as positive quantity, the field of quantity and opera- tion in algebra is greatly extended and some processes are made simpler, others more powerful. These same advan- tages are secured by the use of fractional and negative exponents. We limit the fractional and negative exponents here treated to those whose terms are either positive or negative integers, and com- mensurable; that is, expressible in terms of the unit of quantity used in the given problem.

L Meaning of a Fractional Exponent. So, in general, af X af X af X af to y factors ' j.? Note that in Ex. Vx See Art. Multiply the two results. Which method gives the more accurate result? What two parts are there to every power?

What is the difference between an exponent and a power? Work again Exercise 19 p. Meaning of fhe Exponent Zero, or of fi. Qifrn By cancellation. Transference of Factors in Terms of a Fraction. How many more figures and symbols are there in the first form than. The micron is a small unit of measure equal to one millionth of a meter. Express it as a part of a meter by use of a negative exponent. The length of.

Express this number as 16 in. Give the value of 5o,o, lJ, 5y, -. Express 40 as some power of 4 divided by itself. Express 2P as some power of x divided by itself. State the value of Simplify the following by performing the indicated operations, and reducing the results: How many of the examples in this Exercise can you work at sight? How many examples in Exercise 12 p. By a numerical substitution, show that — does not equal a?. Practice oral work as in Exercise 30 p.

Folynomials whose Terms eontain Fractional or Vegatiye Exponents. Multiply : 2. How many times greater? Illustrate numerically. Also between - 2 -2 and - Give a numerical illustration. Illustrate by giving a, b, X, and y convenient numerical values. Explain as in Ex. Make up an example similar to each of the three preceding examples. In the year the record time for 1 mile traveled on a bicycle was 1 min. Find the latter record. How many pounds of each grade of coffee were used?

Who first suggested the use of a fractional exponent and when? Who first showed that such exponents could be used accord- ing to mathematical laws? Who first used zero as an exponent and when? Give the value of each of the following :. Give the reciprocal of 2. Indicated Soots. For some purposes, one of these methods is better ; for some, the other method. But if we have 7V'12 — 6V3 -f 2Vi8, where exponents are alike, but coefficients and bases unlike, it is usually better to use the radical sign to indicate roots.

In the preceding chapter we considered exponents ; we have now to investigate the properties of radicals. The radicand is the quantity under the radical sign. In treating radicals, we deal only with principal roots see Art. An indicated root which may be exactly ex- tracted is said to he rational; as V5i7, since the cube root of 27 is 3.

A surd is an indicated root which cannot be exactly ex- tracted ; as V5, V4. The Coefficient of a radical is the number prefixed to the radical proper, to show how many times the radical is taken. Entire Surds. If a surd has unity for its coefficient, it is said to be entire. The Degree of a radical is the number of the indicated root. Similar Badioals are those which have the same quantity under the radical sign and the same index. The coefficients and signs of the radicals may be unlike.

Hence, similar radicals must be alike in two respects, and may be unlike in two other respects. Thus, 3 V7 and — 5V7 are similar radicals. Fundamental Principle. Since a radical and a quantity affected by a fractional exponent differ only in form, in investigating the properties of radicals we may use the properties obtained for fractional exponents.

Simplification of a Quantity under the Eadical Sign. V72 X 40 V3f Also that of VIy2. Who first used the sign V to denote a root? Vl2 5. Making Entire Surds. Simplification of Indices. Beducing Radicals to the Same Index. Reduce to equivalent radicals of the same lowest degree : How many examples in Exercise 31 p. Operations with Radicals Addition of Badicah. Hnltiplieation and Involution of Radicals. Division of Radicals. Then simplify first and compute.

Com- pare the amount of work in the two processes. State some of the advantages in being able to sim- plify radicals. Va6 by Vbc Expand : V 8 ViTy-Va;-y 2 State the rule for multiplying one radical by another. Also one similar to Ex. How many examples in Exercise 32 p. Sationalizing a Denominator. State the rule for rationalizing a monomial denom- inator.

State the rule for rationalizing a binomial denom- inator. Use the process of rationalizing the denominator as an aid in finding to four decimal places the numerical value 'x9. Square Root of a Binomial Surd If the surds are dissimilar, one of them must have under the radical sign a factor which the other does not contain.

This factor must remain under the radical sign in the product. The sum or the difference of two dissimilar quad- ratic surds cannot equal a rational quantity. I8-I2V2 The term "imaginary" is used because, so long as we confine ourselves to plus quantity and to its direct opposite, minus quantity, there is no number which mul- tiplied by itself will give a negative number, as — 4, for instance.

All the quantity considered hitherto, that is, all positive or negative quantity, whether it is rational or irrational, is called real quantity. If we extend the realm of quantity outside of positive and negative quantity, imaginary numbers are as real as any others, as will be shown in the next article. Hence, V— 1 on our figure will be represented by OB.

We thus perceive that the introduction of imaginary quantity enlarges the field of quantity considered in algebra from mere quan- tity in a line to quantity in a plane. This gives a vast extension to the power of algebraic processes and introduces many economies in them, as will be found by the student who pursues the study of mathematics extensively. In taking up the subject for the first time, we consider only a few of the first properties of iraaginaries, so called. But if we extract the square root of — 1, we shall have V— 1.

Hence, we must limit the product V— 1 x V— 1 to — 1. Powers of V— 1. The same fact is clear from the figure in Art. Operations with Imaginaries. It follows from Art. But in dividing first indicate the division and afterwards rationalize the denominator, Ex. Root Let the pupil work the above examples using i instead of — 1. Express in terms of i the results obtained in Exs. Multiply : 8. Multiply a; - -by a? SV-a'by -2Va For what values of a; is V2 — a; imaginary?

Simplify : Who first discussed imaginary quantities and when? Who first put the use of these quantities on a scientific basis? Rationalize the denominator of Reduce to simplest form : 2. What advantage is it to know the principle contained in Ex. That contained in Ex. Veed and Utility of Equations of the Second Degree. A certain field of grain is 60 rd. A Quadratic Equation of one unknown quantity is an equation containing the second power of the unknown quantity, but no higher power.

A pure quadratic equation is one in which the second power of the unknown quantity occurs, but not the first power. A pure quadratic equation is sometimes termed an incomplete quadratic equation. An affected or complete quadratic equation is one in which both the first and second powers of the unknown quantity occur. Solution of Pure Qnadratioe. Since only the second power, a? These two values of x are best written together.

Roots, Check. State Ex. A certain iSoor is to be four times as long as it is wide and is to contain 10, sq. Find its dimensions. The width of a certain field is one fourth of the length. If each side of the field is increased by one fourth of itself, the area of the field is increased by 22, sq. Find the dimensions of the field. Who first formed the idea of absolute or indepen- dent negative numbers?

How was nega- tive number used before this? How did the Arabs treat it? Affected Quadratic Equations Completing the Square. An affected quadratic equa- tion may in every instance be reduced to the form a? An equation in this form may then be solved by a process called completing the square.

The use of familiar elementary processes then gives the value of x. Before clearing an equation of f ractions, it is important to reduce each fraction to its simplest form. Let the pupil give the correct solution of Ex. The square of a certain number diminished by 5 times the number equals 6.

Find three consecutive numbers the sum of whose squares is Literal Quadratic Equations are solved by the methods employed in solving quadratic equations with numerical coeflBcients. The factorial method of solution is especially helpful in solving certain literal quadratic equations.

Solve p -f 3 a? This example is the same as Ex. On comparing the two solutions, we observe that at least three fourths of the labor of solution is saved by use of the factorial method. Also of -Wl. Solve a? Also solve the same equation by the factorial method.

Compare the amount of work in the two proc- esses. Why do we not solve all quadratic equations by the factorial method? Solve by the factorial method : An example similar to Ex. How many examples in this Exercise can you solve at sight? Equations in the Quadratic Form Simple Unknown Quantity. An equation containing only two powers of the unknown quantity, the index of one power being twice the index of the other power, is an equation of the quadratic form.

It may be solved by the methods already given for affected quadratic equations. This equation might also have been solved by the fac- torial method. Componnd Unknown Quantity. A polynomial may be used in place of a single quantity as an unknown quantity. Radical Equations If an eiiuation is cleared of radicals, the result is often a quadratic equation. See the treatment of extraneous roots, pp. State the rule for solving radical equations.

Similarly, state Ex. Practice oral work with exponents as in Exercise 89 p. Other Methods of Solving Qnadratie Equations, besides those given in the preceding part of this chapter, may be used. One of these methods may occasionally be used to advantage for some special purpose. After simplifying the equation. Multiply through by four times the coefficient of 2? Use of Formula. Any quadratic equation can be reduced to the form aaj2 4.

Solving this equation by use of Art. Solve 8 aj? Substituting for a, 6, c in the formula on page , Let the pupil check the work. Vx-1 4. Who, as far as we know, first solved a quadratic equation, and at about what time?

How have the different cases in the solution of a quadratic equation been classified at different tiroes? One number exceeds another by 4, and the sum of the squares of the two numbers is The sum of the squares of three consecutive num- bers is Three times the square of a given number, dimin- ished by twice the product of the number and the next lower number, gives The denominator of a given fraction exceeds the numerator by 3, and the sum of the fraction and its reciprocal is 2A.

Find the fraction. The length of a given rectangle exceeds the width by 6 yd. Find the dimensions of the rectangle. The base of a triangle is 4 ft. Find the base of the triangle. The lot is twice as long as it is wide. If the area of the lot and driveway together is sq. A farmer has a field 60 rd. The box is to be 10 in. Find the length of a side of the square sheet of tin.

One baseball nine has won 6 games out of 15, and another has won 8 out of How many straight games must the first nine win from the second, in order that the average of games won by the nines shall be the sanus? A and B together can dp a given piece of work in 2 days. Working alone, A could do the work in 8 days less than B.

How many days would it take each man working alone? How long does it take each pipe alone to fill it? Two boats raced 24 miles. The first boat traveled 8 miles an hour, and the second traveled 14 miles at a Q certain rate and then increased its speed 4 miles an hour. The second boat lost by 20 minutes. Find the rates at which the second boat traveled. A boatman can row 16 miles down a stream and back in 6 hours. If the rate of the stream is 2 miles an hour, find the rate of the boatman in still water.

In a number containing two digits, the left-hand digit is four times as large as the right-hand digit. The product of that number and the number obtained by inverting the digits is Find the original number. In a given number, the tens' digit is one third of the units' digit, and the product of the number obtained by inverting the digits is In a given number, the units' digit exceeds the tens' digit by 3, and the product of that number by the number obtained by inverting the digits is The side of a given square is IJ ft.

By how much must this side be increased, in order that the area of the square may be increased by sq. Two men, A and B, can together do a piece of work in 12 days. B would need 10 days more than A to do the whole work. A given field of grain contains 20 acres and is twice as long as it is wide.

How wide a strip in rods must be cut around the field, in order that of the grain shall be left uncut? A coal bin is to be 8 ft. If 40 cu. One leg of a given triangle exceeds the other by 2 ft. If the hypotenuse is 10 ft.

Find the side of a square whose diagonal is a inches. The product of two consecutive numbers is b. The side of a given square is a feet. By how many feet must this side be increased, in order that the area of the square may be increased by h sq. To two other examples in this Exercise which you think are interesting or suggestive. Find, in feet, the length of a tether by which a cow must be tied in order that she may graze over one half of an acre.

What is the meaning of the above formula in Solid Geometry? What is the meaning of this problem in Solid Geometry? What is the meaning of this problem when applied to a falling body? As applied to projectiles, what is the meaning of the problem? Explain the meaning of the probletn as applied to projectiles.

If an arrow shot over the top of a tower reaches the ground in 6J seconds, determine the height of the steeple resistance of the air being neglected. Is the actual height of the steeple greater or less than the height as thus calculated? The area of a rectangular building lot is to be sq. Find the dimensions of the lot.

Try to solve the problem by use of only one unknown, as X, Even if you succeed in getting a solution, you will find the method awkward and inconvenient. Quadratic Equations containing Two Unknowns. By giving a, J, c, etc. Thus, equations 1 and 2 in Art. In general, the combination of two simultaneous quad- ratic equations by elimination gives an equation of the fourth degree in one unknown, which cannot be solved by the methods taught in this book.

Two simultaneous quadratic equations can be solved by elementary methods only in certain special cases. A Homogeneous Equation is one in which all the terms containing an unknown quantity are of the same degree. General Methods of Solution Cask I When One Equation is of the First Degree and the Other of the Second, two simultaneous equations may always be solved by the method of substitution. J Check. The sum of two numbers is 3, and twice the square of the second number diminished by five times the square of the first number gives 3.

Find the two numbers. How many examples in Exercise 42 p. Case II Solve 2a? J Let the pupil check the work. Two simultaneous equations of the kind treated in Case II may also be solved by eliminating the absolute term between them and factoring to find the value of one unknown in terms of the other, and then proceeding as in Case I.

Point out the examples in Exercise 66 p. Under Case II. Point out the homogeneous equations in Exs. Express Ex. Work the example of Art. Find a number consisting of two digits such that if the number is multiplied by the left-hand digit, the result will be ; but if the number is multiplied by the right- hand digit, the result will be Make up but do not solve an example in each of the cases studied thus far in this chapter.

How many examples in Exercise 44 p. The methods of Cases I and II are the only general methods which can be used in solving all simultaneous quadratic equations of a given class. Besides these, how- ever, there are certain special methods which enable us to solve important particular examples- Examples which come directly under Cases I and II are often solved more advantageously by one of these special methods. The special methods apply with particular advantage to symmetrical equations.

A Symmetrical Equation is one in which, if y is sub- stituted for a:, and x for y, the resulting equation is iden- tical with the original equation. Addition and Snbtraotion Method often in connection with multiplication and division.

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I think it's too early to jump aboard the Axovant Sciences train. While it's great news that the company now has a clear strategy for moving forward, there's still a long way to go. Still, I'd prefer to wait and see what happens with the drug in its late-stage clinical study. That leaves TherapeuticsMD.

If approved, TXHR could get close to that level within five years. So is TherapeuticsMD a slam-dunk pick? I wouldn't go that far. It could take Imvexxy over a decade to reach peak sales. TXHR might have challenges gaining reimbursement because it's a combination of two drugs that are already available in generic form. I think there's a good chance that TherapeuticsMD stock will move a lot higher over the next few years.

My take, though, is to see how the initial launches of Imvexxy and TXHR go -- assuming it wins approval. Investing Best Accounts. Stock Market Basics. Stock Market. Industries to Invest In. Getting Started. Planning for Retirement. Retired: What Now? Personal Finance. The Ascent. About Us. Who Is the Motley Fool?

When we think about risks of transactions, we certainly evaluate kind of the FTC-related risk at times. But I will say that we have had experience with those types of analysis when we've done acquisitions throughout our history. So I think we understand where the risks are around that.

And having said that, I feel like, at least my view is is we continue to have opportunities that will not be limited by that risk. I do think that we have been a company that does not necessarily feel like we need to grow just for growth's sake by making acquisitions.

We feel like we want to do solid deals that are good long-term value for our shareholders that have strategic rationale for our company, and therefore we'll wait for the right opportunities to come our way. And I think that's how you've seen us play this out over the last couple of years.

I don't know if I've hit all of the aspects of your question, I'm certainly willing to go back and cover any of those, or I don't know if Keith wants to add anything to that. But I think very high level, I'm trying to address everything you brought up in your question.

Yes, I think for the most part you certainly did. I don't know, if Keith wants to interject that would be great. I think Josh covered it well. We continue to have an appetite to grow. I think one of our core competencies is growing through acquisitions.

As Josh said, we have a pretty good model in terms of how to deliver on synergies as we work through this. We understand the risks, and I don't think we are limited, whether it be here in Nevada or other places, in order to continue to grow. So we'll continue to look for good opportunities, opportunities that can grow the company. I think Josh made a very important statement, which is we've never been about growing for growth's sake.

We've been about growing to be able to increase profitability and shareholder value and have something that makes strategic sense for us, and we'll continue to operate that way. I think the one other aspect that I would mention is is that absent acquisitions, we feel like we have a lot of opportunity internally to continue on our path of improving our overall operations, continuing to execute on many of the initiatives that we've spoken about over the last several years. So we don't need -- acquisitions are something we will continue to evaluate.

But absent acquisitions, I think we feel like we have a really good foundation of a portfolio that we can manage, that we have things we want to do with that to improve how they operate and how they work together.

And so absent acquisitions, I think we feel comfortable with where the business is and the trajectory of that business. And then if I just could one follow-up as I think about the new guidance and the midpoint of the new guidance. I obviously back off that the that you talked about being in there for the Pinnacle, Valley Forge and Lattner acquisitions, I think you guys had provided enough color on EBITDA through the 3Q of the stuff that you had acquired earlier in the year.

But the underlying statement is correct, Carlo, that basically the guidance we gave you was what was achieved by the acquisitions that we made. And you have to be careful whether you're including Lattner or not including Lattner. Hey guys. Good afternoon.

So, Josh or Keith, if I can start just around the guidance. You guys continue to do a very good job of expanding your margins. And I don't think you'll answer this directly, but can you give us somewhat of an idea in terms of the way you guys are thinking about margin expansion, not only for but also for as well? Or maybe a different way to ask is how you guys are kind of thinking about the flow through opportunities in the near term would be helpful.

And I don't think that is something that we would expect to continue at that level. Las Vegas locals may be a little bit for the higher end of that range. Midwest and south maybe kind of toward the lower end of that range, maybe even a little bit below that at times. We feel very comfortable with that kind of expectation from us. I think our operators do a really good job of running the business along the lines of how we think about it, which is from the perspective of driving profitable revenues, not just trying to drive revenues for growth but driving profitable revenues.

And I think that, Steve, you will continue to see us do that. Many of the tools and capabilities that we are building out will help us to get better at that over time and be more effective at that. So I think as I kind of try to frame the answer for you, this is how we would think about it over the next year or two years. And I don't think we see much in the way of changing that, but just more opportunities to kind of reinforce around that flow through level.

And maybe the follow up on that. Do you guys have any type of internal long-term target in terms of where you think you can get margins? And I'm not looking obviously for the number, but not sure if internally you guys kind of have a goal in your mind?

I think we do. It's certainly higher than it is today. I think what Josh was alluding to is we feel good about the opportunity to continue to see margin expansion in the various parts of our business. I think we've executed on a plan over the last couple of years to grow margins. We'll continue to do that, but some of the tools that are rolling out are still in their infancy and will continue to kind of gain in sophistication and gain traction. So we feel good about being able to continue to expand margins kind of across the portfolio.

And then last question, and maybe I'm overthinking this a little bit too much and I know it's still early on, but you've obviously had competitors with Palace Station and Palms put a lot of money into those assets. And I guess the question is around whether it's the New Orleans or maybe even Cannery and Aliante, but wanted to see if you've seen any impact at those assets to those assets? And then maybe also if you've seen any uptick in promotional spending in or around the market. Well, a couple of comments.

First, look, we're very happy with the kind of condition of the Las Vegas locals market. It's extremely healthy, and you're seeing kind of good revenue growth in the market. We're even more happy or more pleased with the performance of our own business. You look at The Orleans, a record fourth quarter. So we're very happy with kind of how we're executing the business. From a promotional aspect, is it elevated? No, I think it generally is -- generally speaking, I think a fairly normal promotional environment.

I mean, there may be a little bit of elevated advertising related to Palms or Palace here and there, but that's just launching of a new product. We're happy with our business at the Gold Coast. And once again, we're happy with the condition of the market. Hey, congrats on the good results. When we think about your EBITDA guidance, can you just give us a little bit more color on how you're thinking about growth by your three segments?

Thomas, I think I'm going to try to stay away from answering that question. I think we give global guidance, and that's what we're comfortable giving. And so let me just leave it at that. OK, it was worth the shot. Can you talk a little bit about the weather impact that you're seeing here today? Anyway to quantify it? Not yet, Thomas. I mean, there's been obviously -- any number of weekends, as anybody follows along what's happening with the weather, up to and including yesterday and today here in Las Vegas where we're seeing unprecedented snow and cold and the like.

And so we all know that in the business that after some of these weather events, there's a little bit of pent-up demand and some of the business comes back. And so we're not prepared today to give any sort of an impact in terms of maybe the first six weeks of the year. It certainly will be a number, but we're going to wait and see how the rest of the quarter turns out before we start to quantify and see how much of the business maybe comes back in the form of pent-up demand before we quantify it.

Hi, thanks. So you guys have been consistently beating at Blue Chip. I think in terms of any -- I think the way to think about it, or at least the way we think about it is Blue Chip did a really good job of kind of managing to the competition that it had. The impact was not what we expected it to be, but I think we have felt largely the impact that we are going to see.

And what we're doing now is having a business that has the opportunity to grow off of the levels that it's at. To the extent of competitive dynamics today essentially as they are today, we would expect Blue Chip to kind of go off this base going into In fairness, just to answer your question, look. We saw the impact that we expected to see out of the battleground markets between the new competitions in Blue Chip.

We saw really significant declines. What the team was able to do, we talked about this in prior quarters, was really grow the business in other areas that worked between them and the new competition that came out of Chicago and other places. And so the team did a remarkable job of being able to offset those declines by picking up business from other places. So as Josh said, I think we feel good about, we're at a stable base now. The impact of the new competition has been felt, whatever it is, and we can grow off of this base at modest levels.

Great, that's [Inaudible] helpful. So is that still the plan? Can you update the magnitude or the timing? Or is that is what it is? Look, I think we feel very comfortable with the levels of synergies that we've outlined when we announced those transactions.

And I would say most, if not all, most of those synergies were expected to be achieved in And that's part of what the -— that's incorporated into the guidance that we've provided. So I would say everything -- there have been really no surprises to the downside. And I think probably some slight surprises to the upside and things that we have learned that the properties that we're acquiring do better than Boyd, and we plan to kind of roll those out through Boyd and vice versa.

So I think we're trying to be -- really adhere to trying to get the best practices of all the assets not only that we run but that we've acquired. And so that's what creates more upside as we go through time, as we identify those. OK, thank you. And then just the last one and I want to address this to quantify it more than was in the prior question.

I think you had given a goal of improving EBITDA margins by basis points to basis points from about to That was something you gave a few years ago. So you're kind of well into that goal with really two years to go. So I'm just wondering, is that to something that you're still shooting for? Or just given the successes that you've had with a lot of your initiatives and everything, is that something that could be adjusted higher? I think the way I would think about it is we're going to achieve our goal first before we set a new goal.

And while I think we feel good about continuing to make progress toward that goal, and we feel good about the initiatives that we have under way that are like Keith described, that are kind of in their infancy, I think we want to kind of start to get the full benefit of those before we kind of see where we're headed from there.

But we feel good about everything that we've been communicating to the investment community over the last several years in terms of the initiatives, the margin improvement. You're seeing it kind of play out generally. And look, at the end of the day, it won't -— not everything is always going to work out.

So there will be sometimes, no doubt, that we don't get the margin improvement that we expected or the flow through. But I would say, generally, that's the general trend and trajectory of our business over a longer period of time.

But it does seem pretty realistic that you could get to the low end of that goal by the end of this year, correct? But just in the prepared remarks, and I might have missed it, did you give a cash interest expense number? I'm just trying to square up the free cash flow number that you provided. So if you come somewhere between and , you subtract to 10, you'll get a cash number. Thank you for repeating that. And then second question would just be in another release, it was also tonight, there was some comment about a little bit of elevated promotional activity in some regions.

I mean, it seems like in terms of what you saw in your core regional business this quarter, it's probably pretty hard to see that. But just curious, have you seen any promotional change or anything that you guys wanted to call out? We hear of activity kind of picking up here and there, but just curious if there's any pattern in what you guys have seen. No, I don't think we've seen any sustained elevated promotional activity in any of the markets, certainly not here in Las Vegas.

I mentioned a little bit earlier, maybe just a little bit around the launch of Palace Station or the relaunch of Palace, all pretty normal. So I wouldn't really call it elevated and nothing that's sustained. As I'm fond of saying, look, people go out from time to time and get a little aggressive for a week or a month, but there's nothing on a sustained basis in any of our markets that it was worth calling out. Shaun, I think we continue to be focused, and I think you'll hear from many of our peers just in terms of just being more effective and efficient with marketing spend and market reinvestment generally.

And we don't really see that changing. That seems to be a philosophy that's pretty widespread within our industry. Hey, guys. Two quick questions. Or is it too early to tell? Well, if you look at kind of sports betting as a whole, not simply as the profitability of that what I will call the one vertical of sports betting, yes it is clearly EBITDA positive.

It's grown new customers, it has customers staying on the property longer, maybe spending a little more money, some uptick in food and beverage here and there. But it really, and I encourage people to think about this not as the profitability of the operation of the sports book but the impact of the overall property.

All right. And then going back to the first question, and sorry I apologize I've been bouncing around various calls, I don't know if you bought any stock back in the fourth quarter? Yes, Harry, we had that in our prepared remarks. So there's nothing keeping you obviously out of the market. You really haven't been blacked out? Wanted to start with Lake Charles. Just curious if you're seeing or expecting upside at Delta Downs given there's pretty substantial roadwork closer to Lake Charles.

No, we're certainly aware of the potential impact of the roadwork. The roads are being impacted for better than a year now with all the construction activity, but just generally in the area we haven't seen anything significant happened to the business to date from the actual road construction itself.

We'll be prepared if it does, but we haven't seen anything to date. And then just one more for me. I'm curious, does this alter your strategy at all or maybe make you pause at all? Just curious of any thoughts on that? Certainly it got our attention, and we looked at it and we're studying it and we're watching what is going on around that.

But as long as sports betting occurs within the four walls of the borders of a state, it really doesn't impact anything. And so it doesn't, in that respect, alter our plans to move forward in Pennsylvania or moving forward in other states. Look, there's any number of states who continue or are continuing to pursue sports betting legislatively, and I believe that that will continue. So it hasn't slowed down the legislative action around sports betting. So yes, it really hasn't kind of changed our focus of, or frankly our excitement about the opportunity to grow that business.

And there actually is another questioner today, that's going to be David Katz with Jefferies. Hi gentlemen. Congrats on the quarter. So look, I just wanted to clarify and drill a little deeper on one matter.

If we're getting to the leverage target that is between four and five times, which I assume does include the leases on a lease-adjusted basis, can we think about other sort of internal capital allocation choices? Or would you consider any acquisitions within the next year or so that push the leverage back up again?

Are those inside or outside the boundary just as we think about what to do with our forecast this year or next? So I think as we think about our leverage goal if you will, our leverage target, and we've had this philosophy for a while, is we remain flexible in terms of when and how we get there, if you will.

So if there is a very strategic acquisition that is priced right and will create long-term value for the company and for our shareholders, then we will go ahead and make that acquisition even if it takes our leverage up a little bit, as long as we can see a path to get back to that four to five times.

And so we ended the year just a tad above five times. But we're confident that next year we'll get to between four and five times, next year being to be clear. So we'll be flexible. We will let it go up if there is a great acquisition out there otherwise we will focus on being between four and five times. And with respect to kind of the lease adjusted leverage, obviously it'll be at a higher end of the four to five than the lower end.

And this will conclude our question-and-answer session. I would now like to turn the conference back over to Josh Hirsberg for any closing remarks. Thanks, William, and thanks everyone for joining the call today. We appreciate all the questions. And if there's any follow up or any other additional questions, feel free to reach out to the company and we'll make ourselves available.

Thank you very much. More BYD analysis. This article is a transcript of this conference call produced for The Motley Fool. While we strive for our Foolish Best, there may be errors, omissions, or inaccuracies in this transcript. As with all our articles, The Motley Fool does not assume any responsibility for your use of this content, and we strongly encourage you to do your own research, including listening to the call yourself and reading the company's SEC filings.

Please see our Terms and Conditions for additional details, including our Obligatory Capitalized Disclaimers of Liability. Motley Fool Transcribing has no position in any of the stocks mentioned. The Motley Fool has no position in any of the stocks mentioned. The Motley Fool has a disclosure policy. The stock market is looking robust, but also showing some signs of excessive bullishness.

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So yes, it really hasn't a little bit for the that you're seeing here today. I *axo rc6 stifel investments* we've executed on no doubt, that we don't promotional spending in or around. And so we all know alluding to is we money trader good about the opportunity to more color on how you're sports book but the impact our business. I think what Josh was see much in the way think about what to do Palace Station or the relaunch of Palace, all pretty normal. So I think as I a little bit of elevated margins by basis points to basis points from about to of pent-up demand and some. And so let me just. But we're confident that next year we'll get to between Downs given there's pretty substantial. Or just given the successes of the day, it won't four and five times, next year being to be clear. Well, if you look at to be focused, and I a whole, not simply as Blue Chip to kind of will continue to kind of any impact at those assets. Just curious of any thoughts up the free cash flow.