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2 Section 7. Quadratic Equations 477 Use the Zero-Product Property and set each factor equal to zero. x - = 0 or x + 1 = 0 x = or x = - 1 x = The solution set is e -1, f. Figure (a) (b) 5 (c) Now Work p r o b l e m 1 When the left side factors into two linear equations with the same solution, the quadratic equation is said to have a repeated solution. This solution is also called a root of multiplicity, or a double root. Ex a mpl e Solving a Quadratic Equation y bfactoring and by Graphing Solve the equation: 9x + 1 = 6x A lgebraic Solution Put the equation in standard form by adding - 6x to each side. 9x + 1 = 6x 9x - 6x + 1 = 0 Factor the left side of the equation. (x - 1)(x - 1) = 0 x - 1 = 0 or x - 1 = 0 Zero-Product Property x = 1 or x = 1 Graphing Solution Graph Y 1 = 9x + 1 and Y = 6x. See Figure 9. Using INTERSECT, the only point of intersection is (0., ), so the solution of the equation is x = 0., rounded to two decimal places. The solution set is This solution is approximate. Figure 9 10 The equation has only the repeated solution 1. The solution set is e 1 f. Now Work p r o b l e m

3 478 CHAPTER 7 Graphs, Equations, and Inequalities Solve Quadratic Equations Using the Square Root Method Suppose that we wish to solve the quadratic equation x = p () where p is a nonnegative number. Proceed as in the earlier examples: x - p = 0 1x - 1p 1x + 1p = 0 x = 1p or x = - 1p Put in standard form. Factor (over the real numbers). Solve. We have the following result: If x = p and p Ú 0, then x = 1p or x = - 1p. () When statement () is used, it is called the Square Root Method. In statement (), note that if p 7 0 the equation x = p has two solutions, x = 1p and x = - 1p. We usually abbreviate these solutions as x = { 1p, read as x equals plus or minus the square root of p. For example, the two solutions of the equation are and, since 4 =, we have The solution set is 5 -, 6. x = 4 x = { 4 x = { Ex a mpl e Solution Solving Quadratic Equations y busing the Square R oot Method Solve each equation: (a) x = 5 (b) 1x - = 16 (a) x = 5 x = { 5 Use the Square Root Method. x = 5 or x = - 5 The solution set is 5-5, 56. (b) 1x - = 16 Use the Square Root Method. x - = { 16 x - = 16 or x - = - 16 x - = 4 x - = -4 x = 6 x = - The solution set is 5 -, 66. Check: Verify the solutions using a graphing utility. Are the solutions provided by the utility exact? Now Work p r o b l e m

6 Section 7. Quadratic Equations 481 Ex a mpl e 5 Solving a Quadratic Equation y busing the Quadratic Formula and by Graphing Find the real solutions, if any, of the equation x - 5x + 1 = 0. A lgebraic Solution The equation is in standard form, so we compare it to ax + bx + c = 0 to find a, b, and c. x - 5x + 1 = 0 ax + bx + c = 0 a =, b = -5, c = 1 With a =, b = -5, and c = 1, evaluate the discriminant b - 4ac. b - 4ac = = 5-1 = 1 Graphing Solution Figure 0 shows the graph of the equation Y 1 = x - 5x + 1 There are two x-intercepts: one between 0 and 1, the other between 1 and. Using ZERO (or ROOT), we find the solutions to the equation are 0. and 1.4, rounded to two decimal places. These solutions are approximate. Since b - 4ac 7 0, there are two real solutions. Use the quadratic formula with a =, b = -5, c = 1, and b - 4ac = 1. x = -b { b - 4ac = { 11 1 = 5 { 11 6 The solution set is e 5-1, f. These solutions are exact. 6 6 Figure Now Work p r o b l e m 4 Ex a mpl e 6 Solving a Quadratic Equation y busing the Quadratic Formula and by Graphing Find the real solutions, if any, of the equation x + = 4x. A lgebraic Solution The equation, as given, is not in standard form. x + = 4x x - 4x + = 0 ax + bx + c = 0 Subtract 4x from both sides to put the equation in standard form. Compare to standard form. With a =, b = -4, and c =, the discriminant is b - 4ac = = 16-4 = -8 Since b - 4ac 6 0, the equation has no real solution. Graphing Solution We use the standard form of the equation and graph Y 1 = x - 4x + See Figure 1. We see that there are no x-intercepts, so the equation has no real solution, as expected based on the value of the discriminant. Figure Now Work p r o b l e m 4 9

7 48 CHAPTER 7 Graphs, Equations, and Inequalities SUMMARY Procedure for Solving a Quadratic Equation Algebraically To solve a quadratic equation algebraically, first put it in standard form: Then: Step 1: Identify a, b, and c. Step : Evaluate the discriminant, b - 4ac. ax + bx + c = 0 Step : (a) If the discriminant is negative, the equation has no real solution. (b) If the discriminant is zero, the equation has one real solution, a repeated root. (c) If the discriminant is positive, the equation has two distinct real solutions. If you can easily spot factors, use the factoring method to solve the equation. Otherwise, use the quadratic formula or the method of completing the square. 5 Solve Problems That Can Be Modeled by Quadratic Equations Many applied problems require the solution of a quadratic equation. Let s look at one that you will probably see again in a slightly different form if you study calculus. Ex a mpl e 7 Solution Constructing a Box From each corner of a square piece of sheet metal, remove a square of side 9 centimeters. Turn up the edges to form an open box. If the box is to hold 144 cubic centimeters (cm ), what should be the dimensions of the piece of sheet metal? Use Figure as a guide. We have labeled the length of a side of the square piece of sheet metal, x. the box will be of height 9 centimeters, and its square base will have x - 18 as the length of a side. The volume 1length * width * height of the box is therefore 1x x - 18 ~ 9 = 91x - 18 The Model Figure x cm x 18 x 18 x cm x 18 x 18 Volume 9(x 18)(x 18) Since the volume of the box is to be 144 cm, we have 91x - 18 = 144 1x - 18 = 16 x - 18 = {4 Divide each side by 9. Use the Square Root Method. x = 18 { 4 x = or x = 14

9 484 CHAPTER 7 Graphs, Equations, and Inequalities 8. Quadratic equations are sometimes called -degree equations. 9. True or False Quadratic equations always have two real solutions. 10. True or False A quadratic equation that is in the form ax + bx + c = 0 is said to be in standard form. Skill Building In Problems 11 0, solve each equation by factoring. Verify your solution using a graphing utility. 11. x - 9x = 0 1. x + 4x = 0 1. x - 5 = x - 9 = z + z - 6 = v + 7v + 6 = x - 5x - = x + 5x + = t - 48 = 0 0. y - 50 = 0 1. x1x = 0. x1x + 4 = 1. 4x + 9 = 1x 4. 5x + 16 = 40x 5. 61p - 1 = 5p 6. 1u - 4u + = x - 5 = 6 x 8. x + 1 x = x - + x - x = - x1x - In Problems 1 6, solve each equation by the Square Root Method. Verify your solution using a graphing utility. 1. x = 5. x = 6. 1x - 1 = x + = y + = z - = x + 4 = 4 + x - In Problems 7 4, solve each equation by completing the square. Verify your solution using a graphing utility. 7. x + 4x = 1 8. x - 6x = 1 9. x - 1 x - 16 = x + x - 1 = x + x - 1 = 0 4. x - x - 1 = 0 In Problems 4 66, find the real solutions, if any, of each equation. Use the quadratic formula. Verify your solution using a graphing utility. 4. x - 4x + = x + 4x + = x - 4x - 1 = x + 6x + 1 = x - 5x + = x + 5x + = y - y + = t + t + 1 = x = 1 - x 5. x = 1 - x 5. 4x = 9x 54. 5x = 4x 55. 9t - 6t + 1 = u - 6u + 9 = x x - 1 = x - x - = x - x = x - x = x1x + = 6. x1x + = x - x = x - 1 x = x x x = x x x = 4 In Problems 67 7, use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real solution, or no real solution, without solving the equation. 67. x - 6x + 7 = x + 4x + 7 = x - 0x + 5 = x - 0x + 4 = x + 5x - 8 = 0 7. x - x - 7 = 0 Mixed Practice In Problems 7 88, find the real solutions, if any, of each equation. Use any method. Verify your solution using a graphing utility. 7. x - 5 = x - 6 = x - 8x + 1 = x - 1x + 4 = x - 19x - 15 = x + 7x - 0 = z = 6z 80. = y + 6y 81. (x - 4) + 5 = 0

10 Section 7. Quadratic Equations x + 4 = 8. x + x = x + x = x(x - 1) = - 7x x(x + ) = - x x x - + x + 1 = 7x + 1 x - x x x x - 1 = 4-7x x + x - Applications and Extensions 89. Pythagorean Theorem How many right triangles have a hypotenuse that measures x + meters and legs that measure x - 5 meters and x + 7 meters? What are the dimensions of the triangle(s)? 90. Pythagorean Theorem How many right triangles have a hypotenuse that measures 4x + 5 inches and legs that measure x + 1 inches and x inches? What are the dimensions of the triangle(s)? 91. Dimensions of a Window The area of the opening of a rectangular window is to be 14 square feet. If the length is to be feet more than the width, what are the dimensions? 9. Dimensions of a Window The area of a rectangular window is to be 06 square centimeters. If the length exceeds the width by 1 centimeter, what are the dimensions? 9. Geometry Find the dimensions of a rectangle whose perimeter is 6 meters and whose area is 40 square meters. 98. Physics An object is propelled vertically upward with an initial velocity of 0 meters per second. The distance s (in meters) of the object from the ground after t seconds is s = -4.9t + 0t. (a) When will the object be 15 meters above the ground? (b) When will it strike the ground? (c) Will the object reach a height of 100 meters? 99. Reducing the Size of a Candy Bar A jumbo chocolate bar with a rectangular shape measures 1 centimeters in length, 7 centimeters in width, and centimeters in thickness. Due to escalating costs of cocoa, management decides to reduce the volume of the bar by 10%. To accomplish this reduction, management decides that the new bar should have the same centimeter thickness, but the length and width of each should be reduced an equal number of centimeters. What should be the dimensions of the new candy bar? 94. Watering a Field An adjustable water sprinkler that sprays water in a circular pattern is placed at the center of a square field whose area is 150 square feet (see the figure). What is the shortest radius setting that can be used if the field is to be completely enclosed within the circle? 100. Reducing the Size of a Candy Bar Rework Problem 99 if the reduction is to be 0%. 95. Constructing a Box An open box is to be constructed from a square piece of sheet metal by removing a square of side 1 foot from each corner and turning up the edges. If the box is to hold 4 cubic feet, what should be the dimensions of the sheet metal? 96. Constructing a Box Rework Problem 95 if the piece of sheet metal is a rectangle whose length is twice its width. 97. Physics A ball is thrown vertically upward from the top of a building 96 feet tall with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s = t - 16t. (a) After how many seconds does the ball strike the ground? (b) After how many seconds will the ball pass the top of the building on its way down? 101. Constructing a Border around a Pool A circular pool measures 10 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of inches, how wide will the border be? (1 cubic yard = 7 cubic feet) See the illustration. 10 ft x

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