Section 25 Quadratic Equations and Inequalities


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1 5 Quadratic Equations and Inequalities 5 a bi 6. (a bi)(c di) 6. c di 63. Show that i k, k a natural number. 6. Show that i k i, k a natural number. 65. Show that i and i are square roots of 3 i. 66. Show that 3 i and 3 i are square roots of 5 i. 67. Describe how you could find the square roots of 8 6i without using a graphing utility. What are the square roots of 8 6i? 68. Describe how you could find the square roots of i without using a graphing utility. What are the square roots of i? 69. Let S n i i i 3 i n, n. Describe the possible values of S n. 7. Let T n i i i 6 i n,n. Describe the possible values of T n. Supply the reasons in the proofs for the theorems stated in Problems 7 and 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 3. (c di) (a bi) Reason 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 3. (c di)(a bi) Reason.. 3. Letters z and w are often used as complex variables, where z x yi, w u vi, and x, y, u, and 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 73 8, express each property of conjugates verbally and then prove the property. 73. zz is a real number. 7. z z is a real number. 75. z z if and only if z is real. 76. z z 77. z w z w 78. z w z w 79. zw z w 8. z/w z/ w Section 5 Quadratic Equations and Inequalities Introduction Solution by Factoring Solution by Completing the Square Solution by Quadratic Formula Solving Quadratic Inequalities Introduction In this book we are primarily interested in functions with real number domains and ranges. However, if we want to fully understand the nature of the zeros of a function or the roots of an equation, it is necessary to extend some of the definitions in Section  to include complex numbers. A complex number r is a zero
2 6 LINEAR AND QUADRATIC FUNCTIONS of the function f(x) and a root of the equation f(x) if f(r). As before, if r is a real number, then r is also an x intercept of the graph of f. An imaginary zero can never be an x intercept. If a, b, and c are real numbers, a, then associated with the quadratic function f(x) ax bx c is the quadratic equation ax bx c Explore/Discuss Match the zeros of each function with one of the sets A, B, or C: Function Zeros f(x) x A {} g(x) x B {, } h(x) (x ) C {i, i} Which of these sets of zeros can be found using graphical approximation techniques? Which cannot? A graphing utility can be used to approximate the real roots of an equation, but not the imaginary roots. In this section we will develop algebraic techniques for finding the exact value of the roots of a quadratic equation, real or imaginary. In the process, we will derive the wellknown quadratic formula, another important tool for our mathematical toolbox. Solution by Factoring If ax bx c can be written as the product of two firstdegree 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 introduced in Section A. ZERO PROPERTY If m and n are complex numbers, then m n if and only if m or n (or both) EXAMPLE Solving Quadratic Equations by Factoring Solve by factoring: (A) 6x 9x 7 (B) x 6x 5 (C) x 3x
3 5 Quadratic Equations and Inequalities 7 Solutions (A) 6x 9x 7 (x 7)(3x ) Factor left side. x 7 or 3x x 7 x 3 The solution set is { 3, 7 }. (B) x 6x 5 x 6x 9 Write in standard form. (x 3) Factor left side. x 3 (C) The solution set is {3}. The equation has one root, 3. But since it came from two factors, we call 3 a double root or a root of multiplicity. x 3x x 3x x(x 3) x or x 3 x 3 Solution set: {, 3 } MATCHED PROBLEM Solve by factoring: (A) 3x 7x (B) x x 9 (C) x 5x CAUTION. One side of an equation must be before the zero property can be applied. Thus x 6x 5 (x )(x 5) does not imply that x or x 5. See Example, part B, for the correct solution of this equation.. The equations x 3x and x 3 are not equivalent. The first has solution set {, 3, while the second has solution set { 3 } }. The root x is lost when each member of the first equation is divided by the variable x. See Example, part C, for the correct solution of this equation.
4 8 LINEAR AND QUADRATIC FUNCTIONS Do not divide both members of an equation by an expression containing the variable for which you are solving. You may be dividing by. Remark 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. Solution by Completing the Square Factoring is a specialized method that is very efficient if the factors can be quickly identified. However, not all quadratic equations are easy to factor. We now turn to a more general process that is guaranteed to work in all cases. This process is based on completing the square, discussed in Section 3, and the following square root property: SQUARE ROOT PROPERTY For any complex numbers r and s, if r s, then r s. EXAMPLE Solutions Solution by Completing the Square Use completing the square and the square root property to solve each of the following: (A) (x (B) x 6x (C) x ) 5 x 3 (A) This quadratic expression is already written in standard form. We solve for the squared term and then use the square root property: (x ) 5 (x ) 5 x 5 x 5 5 Apply the square root property. Solve for x. (B) We can speed up the process of completing the square by taking advantage of the fact that we are working with a quadratic equation, not a quadratic expression. x 6x x 6x x 6x 9 9 (x 3) x 3 x 3 Complete the square on the left side, and add the same number to the right side.
5 5 Quadratic Equations and Inequalities 9 (C) x x 3 x x 3 x x 3 x x 3 (x ) Make the leading coefficient by dividing by. Complete the square on the left side and add the same number to the right side. Factor the left side. x x i i Answer in a bi form. MATCHED PROBLEM Solve by completing the square: (A) (x (B) x 8x 3 (C) 3x 3 ) 9 x 3 Explore/Discuss Graph the quadratic functions associated with the three quadratic equations in Example. Approximate the x intercepts of each function and compare with the roots found in Example. Which of these equations has roots that cannot be approximated graphically? Solution by Quadratic Formula Now consider the general quadratic equation with unspecified coefficients: ax bx c a We can solve it by completing the square exactly as we did in Example, part C. To make the leading coefficient, we must multiply both sides of the equation by /a. Thus, x b a x c a Adding c/a to both sides of the equation and then completing the square of the left side, we have x b a b b x a a c a
6 5 LINEAR AND QUADRATIC FUNCTIONS 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 b ac a b b ac a See Problem 77. We have thus derived the wellknown and widely used quadratic formula: THEOREM QUADRATIC FORMULA If ax bx c, a, then x b b ac a The quadratic formula and completing the square are equivalent methods. Either can be used to find the exact value of the roots of any quadratic equation. EXAMPLE 3 Solution Using the Quadratic Formula Solve x 3 x by use of the quadratic formula. Leave the answer in simplest radical form. x 3 x x 3 x x x 3 x b b ac a () () ()(3) () Multiply both sides by. Write in standard form. a, b, c 3 CAUTION. () 6 and () ( )
7 5 Quadratic Equations and Inequalities 5 MATCHED PROBLEM 3 5 Solve x 3x using the quadratic formula. Leave the answer in simplest radical form. Explore/Discuss 3 Given the quadratic function f(x) ax bx c, let D b ac. How many real zeros does f have if (A) D (B) D (C) D In each of these three cases, what type of roots does the quadratic equation f(x) have? The quantity b ac in the quadratic formula is called the discriminant and gives us information about the roots of the corresponding equation and the zeros of the associated quadratic function. This information is summarized in Table. T A B L E Discriminants, Roots, and Zeros Discriminant b ac Positive Negative Roots of* ax bx c Two distinct real roots One real root (a double root) Two imaginary roots, one the conjugate of the other Number of Real Zeros of* f(x) ax bx c *a, b, and c are real numbers with a. EXAMPLE Solution Design A picture frame of uniform width has outer dimensions of inches by 8 inches. How wide (to the nearest tenth of an inch) must the frame be to display an area of square inches? We begin by drawing and labeling a figure: x x 8 x 8 If x is the width of the frame, then x must satisfy the equation (8 x)( x) ()
8 5 LINEAR AND QUADRATIC FUNCTIONS Note that x must satisfy x 6 to insure that both x and 8 x are nonnegative. The roots of this quadratic equation can be found algebraically or approximated graphically. Using both methods will confirm that we have the correct answer. We begin with the algebraic solution: (8 x)( x) 6 36x x x x 6x 76 x 5x 9 x 5 9 FIGURE Thus, the quadratic equation has two solutions (rounded to one decimal place): 5 x and x The first must be discarded as being much too large. So the width of the frame is. inches. Graphing both sides of equation () for x 6 and using an intersection routine confirms that this answer is correct (Fig. ). MATCHED PROBLEM A, square foot garden is enclosed with 5 feet of fencing. Find the dimensions of the garden to the nearest tenth of a foot. Solving Quadratic Inequalities Explore/Discuss Graph f(x) (x )(x 3) and examine the graph to determine the solutions of the following inequalities: (A) f(x) (B) f(x) (C) f(x) (D) f(x) The simplest method for solving inequalities involving a function is to find the zeros of the function and then examine the graph to determine where the function is positive and where it is negative. Inequalities involving quadratic functions are handled routinely by this method, as the following examples illustrate. EXAMPLE 5 Solution Finding the Domain of a Function Find the domain of f(x) x x 9. Express answer in interval notation using exact values. The domain of this function is the set of all real numbers x that produce real values for f(x) (Section 3).
9 5 Quadratic Equations and Inequalities 53 This is precisely the solution set of the quadratic inequality x x 9 () FIGURE y x x 9. 5 The solution of this inequality consists of all values of x for which the graph of y x x 9 is on or above the x axis. Using either completing the square or the quadratic formula, we find that the x intercepts are x 5 3 Examining the graph in Figure, we see that the solution of inequality () and, hence, the domain of f, is (, 3] [ 3, ) MATCHED PROBLEM 5 Find the domain of g(x) using exact values. x x. Express answer in interval notation EXAMPLE 6 Projectile Motion If a projectile is shot straight upward from the ground with an initial velocity of 6 feet per second, its distance d (in feet) above the ground at the end of t seconds (neglecting air resistance) is given approximately by d(t) 6t 6t (A) What is the domain of d? (B) At what times (to two decimal places) will the projectile be more than feet above the ground? Express answers in inequality notation. Solutions (A) Factoring d(t), we have d(t) 6t 6t 6t( t) Thus, d() and d(). The projectile is released at t seconds and returns to the ground at t seconds, so the domain of d is t. (B) Since we are asked for twodecimalplace accuracy, we can solve this problem graphically. Graph d and the horizontal line y and find the intersection points (Fig. 3). FIGURE (a) (b)
10 5 LINEAR AND QUADRATIC FUNCTIONS From Figure 3 we see that the projectile will be above feet for.6 t 8.5. MATCHED PROBLEM 6 Refer to the projectile equation in Example 6. At what times (to two decimal places) during its flight will the projectile be less than 5 feet above the ground? Express answer in inequality notation. Answers to Matched Problems. (A) {, 5 (B) (C) {, 5 3 } {3 } (a double root) }. (A) x ( )/3 (B) x 9 (C) x (6 i3)/3 or (3/3)i 3. x (3 9)/. 3. ft by 5.9 ft 5. [ 3, 3] 6. t.9 or 8.6 t EXERCISE 5 A In Problems 6, solve by factoring.. u 8u. 3A A 3. 9y y. 6x 8x 5. x x 6. 8 x 3x In Problems 7 8, solve by using the square root property. 7. m 8. y 5 9. x 5. x 6. 9y 6. x 9 3. x 5. 6a 9 5. (n 5) 9 6. (m 3) 5 7. (d 3) 8. (t ) 9 In Problems 9 6, solve using the quadratic formula. 9. x x 3. x 6x 3. x 8 x. y 3 y 3. x x. m 3 6m 5. 5x x 6. 7x 6x In Problems 7 3, solve and graph. Express answers in both inequality and interval notation. 7. x 3x 8. x x 9. x x 3. x 7x 3. x 8x 3. x 6x 33. x 5x 3. x x B In Problems 35, find exact answers and check with a graphing utility, if possible. 35. x 6x y y y 6y d d 39. 3x x. 3x 5x. x 7x. 9x 9x 3. x 3x. x x In Problems 5 8, solve for the indicated variable in terms of the other variables. Use positive square roots only. 5. s for t 6. a b c gt for a 7. P EI RI for I 8. A P( r) for r In Problems 9 5, solve to two decimal places. Express answers in inequality notation. 9..7x 3.79x x.8x x.x x 7.7x.3 In Problems 53 6, find the domain of each function. Express answers in interval notation using exact values. 53. f(x) x 9 5. g(x) x
11 5 Quadratic Equations and Inequalities h(x) x x k(x) 6 3x 7x F(x) 58. G(x) 6x x 8x x 59. Consider the quadratic equation x x c 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 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table. In Problems 6 6, use the given information concerning the roots of the quadratic equation ax bx c, a, to describe the possible solution sets for the indicated inequality. Illustrate your conclusions with specific examples. 6. ax bx c, given distinct real roots r and r with r r. 6. ax bx c, given distinct real roots r and r with r r. 63. ax bx c, given one (double) real root r. 6. ax bx c, given one (double) real root r. 65. Give an example of a quadratic inequality whose solution set is the entire real line. 66. Give an example of a quadratic inequality whose solution set is the empty set. C Solve Problems 67 7 and express answer in a bi form. 67. x 3ix 68. x 7ix 69. x ix 3 7. x ix 3 with (b ac)/a b ac/a 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 (a b) a ab b b ab a (a b) (b a) a b b a a b a b APPLICATIONS 79. Numbers. Find two numbers such that their sum is and their product is. 8. 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 Numbers. Find two consecutive positive integers whose product is Profit Analysis. A screen printer produces custom silkscreen apparel. The cost C(x) of printing x custom Tshirts and the revenue R(x) from the sale of x Tshirts (both in dollars) are given by C(x).5x R(x) x.5x Determine the production levels x (to the nearest integer) that will result in the printer showing a profit. In Problems 7 and 7, find all solutions. 7. x 3 7. x 73. 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, 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 8. Profit Analysis. Refer to Problem 83. Determine the production levels x (to the nearest integer) that will result in the printer showing a profit of at least $ Air Search. A search plane takes off from an airport at 6: A.M. and travels due north at miles per hour. A second
12 56 LINEAR AND QUADRATIC FUNCTIONS plane takes off at 6:3 A.M. and travels due east at 7 miles per hour. The planes carry radios with a maximum range of 5 miles. When (to the nearest minute) will these planes no longer be able to communicate with each other? 86. Projectile Flight. If a projectile is shot straight upward from the ground with an initial velocity of 76 feet per second, its distance d (in feet) above the ground at the end of t seconds (neglecting air resistance) is given approximately by d(t) 76t 6t (A) What is the domain of d? (B) At what times (to two decimal places) will the projectile be more than feet above the ground? Express answers in inequality notation. 87. Construction. A gardener has a 3 foot by foot rectangular plot of ground. She wants to build a brick walkway of uniform width on the border of the plot (see the figure). If the gardener wants to have square feet of ground left for planting, how wide (to two decimal places) should she build the walkway? x (B) Building codes require that this building have a crosssectional area of at least 5, square feet. What are the widths of the buildings that will satisfy the building codes? (C) Can the developer construct a building with a crosssectional area of 5, square feet? What is the maximum crosssectional area of a building constructed in this manner? 9. Architecture. An architect is designing a small Aframe cottage for a resort area. A crosssection of the cottage is an isosceles triangle with a base of 5 meters and an altitude of meters. The front wall of the cottage must accommodate a sliding door positioned as shown in the figure. meters w h DOOR DETAIL Page of feet 3 feet 5 meters 88. Construction. Refer to Problem 87. The gardener buys enough brick to build 6 square feet of walkway. Is this sufficient to build the walkway determined in Problem 87? If not, how wide (to two decimal places) can she build the walkway with these bricks? 89. Architecture. A developer wants to erect a rectangular building on a triangularshaped piece of property that is feet wide and feet long (see the figure). feet REBEKAH DRIVE Property A l Proposed Building FIRST STREET feet Property Line (A) Express the crosssectional area A(w) of the building as a function of the width w and state the domain of this function. [Hint: Use Euclid s theorem* to find a relationship between the length l and width w.] w (A) Express the area A(w) of the door as a function of the width w and state the domain of this function. [See the hint for Problem 89.] (B) A provision of the building code requires that doorways must have an area of at least. square meters. Find the width of the doorways that satisfy this provision. (C) A second provision of the building code requires all doorways to be at least meters high. Discuss the effect of this requirement on the answer to part B. 9. 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. *Euclid s theorem: If two triangles are similar, their corresponding sides are proportional: a c a a c a b b c c b b
13 6 Additional Equation Solving Techniques 57 Factory A Factory B 9. Construction. A mile track for racing stock cars consists of two semicircles connected by parallel straightaways (see the figure). To provide sufficient room for pit crews, emergency vehicles, and spectator parking, the track must enclose an area of, 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.] Warehouse Warehouse Factory A Factory B Warehouse Odometer Readings 586 5??? , square feet Section 6 Additional Equation Solving Techniques Equations Involving Radicals Equations of the Form ax p bx p c In this section we examine equations that can be transformed into quadratic equations by various algebraic manipulations. With proper interpretation, the solutions of the resulting quadratic equations will lead to the solutions of the original equations. Equations Involving Radicals Consider the equation x x () 5 FIGURE y x, y x Graphing both sides of the equation and using an intersection routine shows that x is a solution to the equation (Fig. ). Is it the only solution? There may be other solutions not visible in this viewing window. Or there may be imaginary solutions (remember, graphical approximation applies only to real solutions). To solve this equation algebraically, we square each side of equation () and then proceed to solve the resulting quadratic equation. Thus, x (x ) () x x x x (x )(x ) x,
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