Name Class Date. Solving Quadratic Equations by Using Square Roots Going Deeper

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1 Name Class Date 8-7 Solving Quadratic Equations by Using Square Roots Going Deeper Essential question: How can you solve a quadratic equation using square roots? 1 PREP FOR A-REI.2.4b ENGAGE Understanding Square Roots You know that 2 2 = 4 and (-2) 2 = 4 and that the numbers 2 and -2 are called the square roots of 4. In general, if x 2 = a, then x is a square root of a. Every positive number a has two square roots. This is illustrated in the diagram using the graph of y = x 2 and letting y = a. Notice that one square root of a is positive and is written a, while the other is negative and is written - a. The symbol is called a radical sign, and the number underneath the radical sign is called the radicand. a y = x 2 When the radicand is a perfect square, you can simplify a square root. For instance, because 4 is a perfect square, you can write the square roots of 4 as ± 4 = ±2. When the radicand is not a perfect square, you may still be able to simplify a square root using one of these properties: - a a Product Property of Radicals: For nonnegative a and b, ab = a b. Quotient Property of Radicals: For nonnegative a and positive b, a b = a. b For instance, because 12 has 4 as one of its factors, you can use the product property to write the square roots of 12 as ± 12 = ± 4 3 = ± 4 3 = ± 2 3. REFLECT 1a. Does 0 have any square roots? Why or why not? 1b. Does a negative number have any square roots? Why or why not? 1c. Explain how you would simplify the square roots of 5 4. Chapter Lesson 7

2 2 Solving a quadratic equation algebraically involves isolating the squared expression in the equation. Once you have the equation in the form (x - h) 2 = c, you can use the definition of a square root to write x - h = ± c and finish solving for x. A-REI.2.4b EXAMPLE Solving Quadratic Equations Algebraically Solve each quadratic equation. A 2x 2-7 = 9 Equation to be solved Add 7 to both sides. 2x 2 = 2x 2 = Divide both sides by 2. x 2 = Definition of a square root Simplify the square roots. B -3 (x - 6) = 7 Equation to be solved Subtract 19 from both sides. -3 (x - 6) 2 = -3 (x - 6) 2 = Divide both sides by -3. (x - 6) 2 = x - 6 = x - 6 = x - 6 = Definition of a square root Simplify the square roots. or Add 6 to both sides. REFLECT 2a. How can you check the solutions of a quadratic equation? Chapter Lesson 7

3 3 A-CED.1.1 EXAMPLE Solving a Real-World Problem A person standing on a second-floor balcony drops keys to a friend standing below the balcony. The keys are dropped from a height of 10 feet. The height (in feet) of the keys as they fall is given by the function h(t) = -16 t where t is the time (in seconds) since the keys were dropped. The friend catches the keys at a height of 4 feet. Write and solve an equation to find the elapsed time before the keys are caught. -16 t = Write the equation to be solved. Subtract 10 from both sides. -16 t 2 = -16 t 2 = Divide both sides by -16. t 2 = t = t Express the right side as a decimal. Definition of a square root Use a calculator to approximate the square roots. The elapsed time before the keys are caught is about. REFLECT 3a. Although the equation that you solved has two solutions, one of them is rejected. Why? 3b. The exact positive solution of the equation is t =. Explain how to obtain this 4 result, and show that it gives the same approximate solution. 3c. Suppose the friend decides not to catch the keys and lets them fall to the ground instead. What equation must you solve to find the elapsed time until the keys hit the ground? What is that elapsed time? 6 Chapter Lesson 7

4 PRACTICE 1. Write the square roots of 64 in simplified form. 2. Write the square roots of 32 in simplified form. 3. Write the square roots of 8_ in simplified form Explain why the square roots of 37 cannot be simplified. Solve each quadratic equation. Simplify solutions when possible. 5. x 2 = x 2 = x = x 2 = (x - 5) 2 = (x + 1) 2 = (x - 7) 2 = (x + 3) 2 = (x + 2) 2-4 = (x - 1) = To study how high a ball bounces, students drop the ball from various heights. The function h(t) = -16 t 2 + h 0 gives the height (in feet) of the ball at time t measured in seconds since the ball was dropped from a height h 0. a. The ball is dropped from a height h 0 = 8 feet. Write and solve an equation to find the elapsed time until the ball hits the floor. b. Does doubling the drop height also double the elapsed time until the ball hits the floor? Explain why or why not. c. When dropped from a height h 0 = 16 feet, the ball rebounds to a height of 8 feet and then falls back to the floor. Find the total time for this to happen. (Assume the ball takes the same time to rebound 8 feet as it does to fall 8 feet.) Chapter Lesson 7

5 Name Class Date 8-7 Additional Practice 1. 2 = = 100 =± 81 =± The solutions are and. = ± =± The solutions are and = = = 400 = ± ± = = = = = = = = = = = = = = = The height of a skydiver jumping out of an airplane is given by = How long will it take the skydiver to reach the ground? Round to the nearest tenth of a second. 19. The height of a triangle is twice the length of its base. The area of the triangle is 50 m 2. Find the height and base to the nearest tenth of a meter. 20. The height of an acorn falling out of a tree is given by = If an acorn takes 1 second to fall to the ground. What is the value of? Chapter Lesson 7

6 Problem Solving A furniture maker has designed a bookcase with the proportions shown in the diagram below. Write the correct answer. 1. A customer has requested a bookcase with the two shelves having a total area of 864 square inches. What should equal to meet the customer s specifications? 2. Barnard has a stain on his wall and would like to cover it up with a bookcase. What should equal in order for the back of the bookcase to cover an area of 4800 square inches? 3. Bria would like to display her collection of soap carvings on top of her bookcase. The collection takes up an area of 400 square inches. What should equal for the top of the bookcase to have the correct area? Round your answer to the nearest tenth of an inch. Select the best answer. 5. Carter plans to wallpaper the longest wall in his living room. The wall is twice as long as it is high and has an area of 162 square feet. What is the height of the wall? A 8 feet C 12 feet B 9 feet D 18 feet 7. Trinette cut a square tablecloth into 4 equal pieces that she used to make two pillow covers. The area of the tablecloth was 3600 square inches. What is the side length of each piece Trinette used to make the pillow covers? A 20 inches C 60 inches B 30 inches D 90 inches 4. Eliana would like to cover the side panels with silk. She has 1600 square inches of silk. What should equal so that she can use all of her silk to completely cover the sides? Round your answer to the nearest tenth of an inch. 6. An apple drops off the apple tree from a height of 8 feet. How long does it take the apple to reach the ground? Use the function ( ) = , where is the initial height of a falling object, to find the answer. F 0.5 seconds H 1 second G 0.71 seconds J 2.23 seconds 8. Elton earns dollars per hour at the bookstore. His mother, Evelyn, earns 2 dollars per hour as a career counselor. Twice Evelyn s wage equals $ What is Elton s hourly wage? Round your answer to the nearest cent. F $4.60 H $9.19 G $6.50 J $13.00 Chapter Lesson 7