58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be two arbitrary complex numbers; then: Statement. (a bi) (c di) (a c) (b d)i. (c a) (d b)i. (c di) (a bi) Reason... 66. Theorem: The complex numbers are commutative under multiplication. Proof: Let a bi and c di be two arbitrary complex numbers; then: Statement. (a bi) (c di) (ac bd) (ad bc)i. (ca db) (da cb)i. (c di)(a bi) Reason... Letters z and w are often used as complex variables, where z x yi, w u vi, and x, y, u, v are real numbers. The conjugates of z and w, denoted by z and w, respectively, are given by z x yi and w u vi. In Problems 67 7, express each property of conjugates verbally and then prove the property. 67. zz is a real number. 68. z z is a real number. 69. z z if and only if z is real. 70. z z 7. z w z w 7. z w z w 7. zw z w 7. z/w z/ w SECTION -6 Quadratic Equations and Applications Solution by Factoring Solution by Square Root Solution by Completing the Square Solution by Quadratic Formula Applications The next class of equations we consider are the second-degree polynomial equations in one variable, called quadratic equations. DEFINITION Quadratic Equation A quadratic equation in one variable is any equation that can be written in the form ax bx c 0 a 0 Standard Form where x is a variable and a, b, and c are constants.
-6 Quadratic Equations and Applications 59 Now that we have discussed the complex number system, we will use complex numbers when solving equations. Recall that a solution of an equation is also called a root of the equation. A real number solution of an equation is called a real root, and an imaginary number solution is called an imaginary root. In this section we develop methods for finding all real and imaginary roots of a quadratic equation. Solution by Factoring If ax bx c can be written as the product of two first-degree factors, then the quadratic equation can be quickly and easily solved. The method of solution by factoring rests on the zero property of complex numbers, which is a generalization of the zero property of real numbers reviewed in Section A-. Zero Property If m and n are complex numbers, then m n 0 if and only if m 0 or n 0 (or both) EXAMPLE Solving Quadratic Equations by Factoring Solve by factoring: (A) 6x 9x 7 0 (B) x 6x 5 (C) x x Solutions (A) (B) (C) 6x 9x 7 0 (x 7)(x ) 0 Factor left side. x 7 0 or x 0 x 7 x The solution set is {, 7 }. x 6x 5 x 6x 9 0 Write in standard form. (x ) 0 Factor left side. x The solution set is {}. The equation has one root,. But since it came from two factors, we call a double root. x x 0 x(x ) 0 x 0 x x or Solution set: {0, } x 0 x
60 Equations and Inequalities Matched Problem Solve by factoring: (A) x 7x 0 0 (B) x x 9 0 (C) x 5x CAUTION. One side of an equation must be 0 before the zero property can be applied. Thus x 6x 5 (x )(x 5) does not imply that x or x 5. See Example B for the correct solution of this equation.. The equations x x and x are not equivalent. The first has solution set {0, }, while the second has solution set { }. The root x 0 is lost when each member of the first equation is divided by the variable x. See Example C for the correct solution of this equation. Do not divide both members of an equation by an expression containing the variable for which you are solving. You may be dividing by 0. Solution by Square Root We now turn our attention to quadratic equations that do not have the first-degree term that is, equations of the special form ax c 0 a 0 The method of solution of this special form makes direct use of the square root property: Square Root Property If A C, then A C.
-6 Quadratic Equations and Applications 6 EXPLORE-DISCUSS Determine if each of the following pairs of equations is equivalent or not. Explain your answer. (A) x and x (B) x and x (C) x and x (D) x and x The use of the square root property is illustrated in the next example. Note: It is common practice to represent solutions of quadratic equations informally by the last equation rather than by writing a solution set using set notation. From now on, we will follow this practice unless a particular emphasis is desired. EXAMPLE Using the Square Root Property Solve using the square root property: (A) x 0 (B) x 7 0 (C) (x ) 5 Solutions (A) x 0 x x or 6 Solution set: 6, 6 (B) x 7 0 x 9 x 9 or i Solution set: { i, i} (C) (x ) 5 x 5 x 5 5 Matched Problem Solve using the square root property: (A) x 5 0 (B) x 8 0 (C) (x ) 9
6 Equations and Inequalities EXPLORE-DISCUSS Replace? in each of the following with a number that makes the equation valid. (A) (x ) x x? (B) (x ) x x? (C) (x ) x 6x? (D) (x ) x 8x? Replace? in each of the following with a number that makes the trinomial a perfect square. (E) x 0x? (F) x x? (G) x bx? Solution by Completing the Square The methods of square root and factoring are generally fast when they apply; however, there are equations, such as x 6x 0 (see Example A), that cannot be solved directly by these methods. A more general procedure must be developed to take care of this type of equation for example, the method of completing the square. This method is based on the process of transforming the standard quadratic equation ax bx c 0 into the form (x A) B where A and B are constants. The last equation can easily be solved by using the square root property. But how do we transform the first equation into the second? The following brief discussion provides the key to the process. What number must be added to x bx so that the result is the square of a firstdegree polynomial? There is a simple mechanical rule for finding this number, based on the square of the following binomials: (x m) x mx m (x m) x mx m In either case, we see that the third term on the right is the square of one-half of the coefficient of x in the second term on the right. This observation leads directly to the rule for completing the square. Completing the Square To complete the square of a quadratic of the form x bx, add the square of one-half the coefficient of x; that is, add (b/). Thus, x bx b x b x 5x x bx x 5x 5 x 5
-6 Quadratic Equations and Applications 6 EXAMPLE Completing the Square Complete the square for each of the following: (A) x x (B) x bx Solutions (A) x x x x 9 (x ) (B) x bx x bx b x b 9 Add ; that is,. b Add b ; that is,. Matched Problem Complete the square for each of the following: (A) x 5x (B) x mx It is important to note that the rule for completing the square applies only to quadratic forms in which the coefficient of the second-degree term is. This causes little trouble, however, as you will see. We now solve two equations by the method of completing the square. EXAMPLE Solution by Completing the Square Solve by completing the square: (A) x 6x 0 (B) x x 0 Solutions (A) (B) x 6x 0 x 6x x 6x 9 9 (x ) x x x x 0 x x 0 x x x x Complete the square on the left side, and add the same number to the right side. Make the leading coefficient by dividing by. Complete the square on the left side and add the same number to the right side.
6 Equations and Inequalities (x ) Factor the left side. x x i i Answer in a bi form. Matched Problem Solve by completing the square: (A) x 8x 0 (B) x x 0 Solution by Quadratic Formula Now consider the general quadratic equation with unspecified coefficients: ax bx c 0 a 0 We can solve it by completing the square exactly as we did in Example B. To make the leading coefficient, we must multiply both sides of the equation by /a. Thus, x b a x c a 0 Adding c/a to both sides of the equation and then completing the square of the left side, we have We now factor the left side and solve using the square root property: x b a b ac a x b a b ac a x b a x b a b a c a x b a b ac a See Problem 75 in Exercise -6. b b ac a We have thus derived the well-known and widely used quadratic formula:
-6 Quadratic Equations and Applications 65 Theorem Quadratic Formula If ax bx c 0, a 0, then x b b ac a The quadratic formula should be memorized and used to solve quadratic equations when other methods fail or are more difficult to apply. EXAMPLE 5 Using the Quadratic Formula x by use of the quadratic formula. Leave the answer in simplest rad- Solve x ical form. Solution x x x x x x x 0 b b ac a Multiply both sides by. Write in standard form. ( ) ( ) ()( ) () 0 0 a, b, c 0 CAUTION. ( ) 6 and ( ) 6.. 0 0 0 0 0 0 0 ( 0) 0 5 Matched Problem 5 Solve x x by use of the quadratic formula. Leave the answer in simplest radical form. EXAMPLE 6 Using the Quadratic Formula with a Calculator Solve 5.7x 6.0x.7 0 to two decimal places using a calculator.
66 Equations and Inequalities Solution 5.7x 6.0x.7 0 x 6.0 ( 6.0) (5.7)(.7) (5.7) 0.5, 0.87 Matched Problem 6 Solve.79x 5.07x 7.69 0 to two decimal places using a calculator. We conclude this part of the discussion by noting that b ac in the quadratic formula is called the discriminant and gives us useful information about the corresponding roots as shown in Table. TABLE Discriminant and Roots Discriminant Roots of ax bx c 0 b ac a, b, and c real numbers, a 0 Positive Two distinct real roots 0 One real root (a double root) Negative Two imaginary roots, one the conjugate of the other For example: (A) x x 0 has two real roots, since b ac ( ) ()( ) 0 (B) x x 0 has one real (double) root, since b ac ( ) ()() 0 (C) x x 0 has two imaginary roots, since b ac ( ) ()() 0 Applications We now consider several applications that make use of quadratic equations. First, the strategy for solving word problems, presented earlier in Section -, is repeated below. Strategy for Solving Word Problems. Read the problem carefully several times if necessary that is, until you understand the problem, know what is to be found, and know what is given.
-6 Quadratic Equations and Applications 67. Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully.. If appropriate, draw figures or diagrams and label known and unknown parts.. Look for formulas connecting the known quantities to the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6. Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem not just the equation found in step 5 since a mistake may have been made in setting up the equation in step 5. EXAMPLE 7 Setting Up and Solving a Word Problem The sum of a number and its reciprocal is. Find all such numbers. 6 Solution Let x the number; then: x x 6 (6x)x (6x) x (6x) 6 Multiply both sides by 6x. [Note: x 0.] 6x 6 x 6x x 6 0 (x )(x ) 0 x 0 x or A quadratic equation x 0 x Thus, two such numbers are and. Check 6 6 Matched Problem 7 The sum of two numbers is and their product is. Find the two numbers. [Hint: If one number is x, then the other number is x.]
68 Equations and Inequalities EXAMPLE 8 A Distance Rate Time Problem An excursion boat takes.6 hours longer to go 6 miles up a river than to return. If the rate of the current is miles per hour, what is the rate of the boat in still water? Solution Let 6 6.6 T D, x, x x x R 6(x ) 6(x ).6(x )(x ) 6x 6x.6x 5.6.6x.6 x 96 x 96 The rate in still water is miles per hour. x Rate of boat in still water x Rate downstream x Rate upstream Time Time.6 upstream downstream [Note: 96 must be discarded, since it doesn t make sense in the problem to have a negative rate.] Check Time upstream D R 6.6 Time downstream D R 6.6 Difference of times Matched Problem 8 Two boats travel at right angles to each other after leaving a dock at the same time. One hour later they are 5 miles apart. If one boat travels 5 miles per hour faster than the other, what is the rate of each? [Hint: Use the Pythagorean theorem,* remembering that distance equals rate times time.] a c b *Pythagorean theorem: A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the two shorter sides: c a b.
-6 Quadratic Equations and Applications 69 EXAMPLE 9 A Quantity Rate Time Problem A payroll can be completed in hours by two computers working simultaneously. How many hours are required for each computer to complete the payroll alone if the older model requires hours longer than the newer model? Compute answers to two decimal places. Solution Let x Time for new model to complete the payroll alone x Time for old model to complete the payroll alone Time for both computers to complete the payroll together Then, Part of job completed by new model in hours x () x Rate for new model x Rate for old model x Part of job completed by old model in hours Completes Completes whole job x 0, x x () of the payroll per hour of the payroll per hour x (x ) x x(x ) Multiply both sides by x(x ). x x x x x 5x 0 x x x 9.77 x 5 7 5 7 6.77 5 7.77 is discarded since x cannot be negative. The new model would complete the payroll in 6.77 hours working alone, and the old model would complete the payroll in 9.77 hours working alone. x
70 Equations and Inequalities Check 6.77 () 9.77 ().000 59 ± Note: We do not expect the check to be exact, since we rounded the answers to two decimal places. An exact check would be produced by using x (5 7)/. The latter is left to the reader. Matched Problem 9 Two technicians can complete a mailing in hours when working together. Alone, one can complete the mailing hours faster than the other. How long will it take each person to complete the mailing alone? Compute the answers to two decimal places. Answers to Matched Problems 5 5. (A) x, (B) x (a double root) (C) x 0,. (A) x 5 or 5/ (B) x i (C) x ( )/. (A) x 5 5 5x (x ) (B) x mx (m /) [x (m/)]. (A) x 9 (B) x (6 i )/ or ( /)i 5. x ( 9)/ 6. x.80, 0.98 7. and 8. 5 and 0 miles per hour 9. 5.6 and 7.6 hours EXERCISE -6 A Leave all answers involving radicals in simplified radical form unless otherwise stated. B In Problems 6, solve by factoring.. x 8x. y 5y. t 9 t. s 6s 5. w w 0 6. 6x 9 x In Problems 7 8, solve by using the square root property. 7. m 5 0 8. n 6 0 9. c 9 0 0. d 6 0. y 9 0. 9x 5 0. 5z 0. 6w 7 0 5. (s ) 5 6. (t ) 7. (n ) 8. (m ) In Problems 9 6, solve using the quadratic formula. 9. x x 0 0. y y 7 0. x x 0. y y 0. t 8 6t. 9s s 5. t 6t 6. 9s 7 s In Problems 7, solve by completing the square. 7. x x 0 8. y y 0 9. r 0r 0 0. s 6s 7 0. u 8u 5 0. v 6v 0. w w 0. z 8z 0 In Problems 5 5, solve by any method. 5. x 7x 0 6. 9x 9x 7. (y ) 5 8. (m ) 9. x x 0. x x. 7n n. 8u u 0. 8. x x 5. 6. 0 m 0 m 7. 8. x x x u u. y. y x x x
-6 Quadratic Equations and Applications 7 9. 50. x x x x 9 x x x x x x x 5. u u 5. 7x x In Problems 5 56, solve for the indicated variable in terms of the other variables. Use positive square roots only. 5. s gt for t 5. a b c for a 55. P EI RI for I 56. A P( r) for r Solve Problems 57 60 to two decimal places using a calculator. 57..07x.79x. 0 58. 0.6x.8x.9 0 59..8x.0x.8 0 60. 5.x 7.7x. 0 6. Consider the quadratic equation x x c 0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table. 6. Consider the quadratic equation x x c 0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table. Use the discriminant to determine whether the equations in Problems 6 66 have real solutions. 6. 0.0x 0.0x 0.00 0 6. 0.5x 0.8x 0.00 0 65. 0.0x 0.0x 0.00 0 66. 0.5x 0.8x 0.0 0 C Solve Problems 67 70 and leave answers in simplified radical form (i is the imaginary unit). 67. x 8 x 68. x x 69. x ix 70. x ix In Problems 7 and 7, find all solutions. 7. x 0 7. x 0 7. Can a quadratic equation with rational coefficients have one rational root and one irrational root? Explain. 7. Can a quadratic equation with real coefficients have one real root and one imaginary root? Explain. 75. Show that if r and r are the two roots of ax bx c 0, then r r c/a. 76. For r and r in Problem 75, show that r r b/a. 77. In one stage of the derivation of the quadratic formula, we replaced the expression with What justifies using a in place of a? 78. Find the error in the following proof that two arbitrary numbers are equal to each other: Let a and b be arbitrary numbers such that a b. Then APPLICATIONS (a b) (b a) a b b a a b a b (b ac)/a b ac/a (a b) a ab b b ab a 79. Numbers. Find two numbers such that their sum is and their product is 0. 80. Numbers. Find all numbers with the property that when the number is added to itself the sum is the same as when the number is multiplied by itself. 8. Numbers. Find two consecutive positive even integers whose product is 68. 0 8. Numbers. The sum of a number and its reciprocal is. Find the number. 8. Geometry. If the length and width of a - by -inch rectangle are each increased by the same amount, the area of the new rectangle will be twice that of the original. What are the dimensions of the new rectangle (to two decimal places)? 8. Geometry. Find the base b and height h of a triangle with an area of square feet if its base is feet longer than its height and the formula for area is A bh. 85. Business. If $P are invested at an interest rate r compounded annually, at the end of years the amount will be A P( r). At what interest rate will $,000 increase to $,0 in years? [Note: A $,0 and P $,000.]
7 Equations and Inequalities 86. Economics. In a certain city, the price demand and price supply equations for CDs are p 75,000 q p 0.0005q.5 Demand equation Supply equation where q represents quantity and p represents the price in dollars. Find the equilibrium price. 87. Puzzle. Two planes travel at right angles to each other after leaving the same airport at the same time. One hour later they are 60 miles apart. If one travels 0 miles per hour faster than the other, what is the rate of each? 88. Navigation. A speedboat takes hour longer to go miles up a river than to return. If the boat cruises at 0 miles per hour in still water, what is the rate of the current? 89. Engineering. One pipe can fill a tank in 5 hours less than another. Together they can fill the tank in 5 hours. How long would it take each alone to fill the tank? Compute the answer to two decimal places. 90. Engineering. Two gears rotate so that one completes more revolution per minute than the other. If it takes the smaller gear second less than the larger gear to complete 5 revolution, how many revolutions does each gear make in minute? 9. Physics Engineering. For a car traveling at a speed of v miles per hour, under the best possible conditions the shortest distance d necessary to stop it (including reaction time) is given by the empirical formula d 0.0v.v, where d is measured in feet. Estimate the speed of a car that requires 65 feet to stop in an emergency. 9. Physics Engineering. If a projectile is shot vertically into the air (from the ground) with an initial velocity of 76 feet per second, its distance y (in feet) above the ground t seconds after it is shot is given by y 76t 6t (neglecting air resistance). (A) Find the times when y is 0, and interpret the results physically. (B) Find the times when the projectile is 6 feet off the ground. Compute answers to two decimal places. 9. Construction. A developer wants to erect a rectangular building on a triangular-shaped piece of property that is 00 feet wide and 00 feet long (see the figure). Find the dimensions of the building if its cross-sectional area is 5,000 square feet. [Hint: Use Euclid s theorem* to find a relationship between the length and width of the building.] 9. Architecture. An architect is designing a small A-frame cottage for a resort area. A cross section of the cottage is an isosceles triangle with an area of 98 square feet. The front wall of the cottage must accommodate a sliding door that is 6 feet wide and 8 feet high (see the figure). Find the width and height of the cross section of the cottage. [Recall: The area of a triangle with base b and altitude h is bh/.] 95. Transportation. A delivery truck leaves a warehouse and travels north to factory A. From factory A the truck travels east to factory B and then returns directly to the warehouse (see the figure). The driver recorded the truck s odometer reading at the warehouse at both the beginning and the end of the trip and also at factory B, but forgot to record it at factory A (see the table). The driver does recall that it was further from the warehouse to factory A than it was from factory A to factory B. Since delivery charges are based on distance from the warehouse, the driver needs to know how far factory A is from the warehouse. Find this distance. Factory A 6 feet 8 feet Factory B Warehouse 00 feet 5,000 square feet 00 feet *Euclid s theorem: If two triangles are similar, their corresponding sides are proportional: a b c a b c a a b b c c
-7 Equations Reducible to Quadratic Form 7 Warehouse Factory A Factory B Warehouse Odometer Readings 5 8 6 5??? 5 9 7 5 0 0 enclose an area of 00,000 square feet. Find the length of the straightaways and the diameter of the semicircles to the nearest foot. [Recall: The area A and circumference C of a circle of diameter d are given by A d / and C d.] 96. Construction. A -mile track for racing stock cars consists of two semicircles connected by parallel straightaways (see the figure). In order to provide sufficient room for pit crews, emergency vehicles, and spectator parking, the track must 00,000 square feet SECTION -7 Equations Reducible to Quadratic Form Equations Involving Radicals Equations Involving Rational Exponents Equations Involving Radicals In solving an equation involving a radical like x x it appears that we can remove the radical by squaring each side and then proceed to solve the resulting quadratic equation. Thus, x x 0 (x )(x ) 0 x ( x ) x x x, Now we check these results in the original equation. Check: x x x Check: x x x Thus, is a solution, but is not. These results are a special case of Theorem.