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9-3 Solving Quadratic Equations Vocabulary Review 1. Multiple Choice Toby wants to buy concert tickets. Each ticket costs $45.00. Circle the function rule Toby could use to find the cost c of any number of tickets t. c 5 45t t 5 45c 45 5 ct c 5 45 2. What is the difference between a relation and a function? ________________________________________________________________________ ________________________________________________________________________ Vocabulary Builder root (noun) root Main Idea: A root is a solution of an equation in the form f (x) 5 0. The related function y 5 f (x) has a zero, and the graph of y 5 f (x) has an x-intercept, for each root of the equation. Example: The equation x 2 2 1 5 0 has two roots. So, the function y 5 x 2 2 1 has two zeros: 21 and 1. Use Your Vocabulary Complete each statement with the correct word from the list below. Each word is used exactly once. root roots zero zeros 3. The function y 5 4x 2 2 1 has two zeros. The equation 0 5 4x 2 2 1 has two 9. 4. Each 9 of an equation in the form f (x) 5 0 is a solution of the equation. 5. The function y 5 2x 2 5 has one 9. 6. One of the two 9 of the function y 5 x 2 2 4 is 2. Chapter 9 266 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Related Words: x-intercept, zero, solution Key Concept Standard Form of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 1 bx 1 c 5 0, where a 2 0. This form is called the standard form of a quadratic equation. Examples 9x 2 1 81 5 0 x 2 1 4x 1 16 5 0 7. Cross out the quadratic equations that are NOT in standard form. x 2 2 81 5 0 12x 2 5 6x 1 20 8x 2 2 10 5 25x 8. Draw a line from each quadratic equation in Column A to an equivalent equation in Column B. Column A Column B x 2 1 3x 1 18 5 5x 1 11 2x 2 1 x 1 1 5 0 3x 2 1 x 5 x 2 2 1 9x 2 1 30x 1 25 5 0 (3x 1 5)2 5 0 x 2 2 2x 1 7 5 0 You can solve quadratic equations by a variety of methods. One way to solve a quadratic equation ax 2 1 bx 1 c 5 0 is to graph the related quadratic function y 5 ax 2 1 bx 1 c. The solutions of the equation are the x-intercepts of the related function. Problem 1 Solving by Graphing Got It? What are the solutions of the equation y Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. x 2 2 16 5 0? Use a graph of the related function. 16 9. Graph the related function, y 5 x 2 2 16. 8 x –8 –4 O 4 8 –8 –16 10. Which part of the graph gives you the solutions of the equation? Circle your answer. vertex x-intercept(s) y-intercept(s) axis of symmetry HSM11_A1MC_0903_T91319 11. Complete each sentence. The equation has solutions. The solutions are and , or 4 . 267 Lesson 9-3 Problem 2 Solving Using Square Roots Got It? What are the solutions of m2 2 36 5 0? 12. Follow the steps to find the solutions of m2 2 36 5 0. 1 2 Write the original equation. Isolate m 2 on one side of the equation. m2 5 Find the square roots of each side and simplify. 3 m56 " 56 Got It? What are the solutions of 3x 2 1 15 5 0? 13. Complete the steps to find the solutions of 3x 2 1 15 5 0. 3x 2 1 15 5 0 Write the original equation. 3x 2HSM11_A1MC_0903_T91320 5 Isolate the x2 term on one side. x 2 5 x54 Å Divide each side by 3. Find the square roots of each side. 14. Does 3x 2 1 15 5 0 have a real-number solution? Explain. ________________________________________________________________________ ________________________________________________________________________ Problem 3 Choosing a Reasonable Solution Got It? An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length < that is twice the width w. The height of the tank is 4 ft. The volume of the tank is 500 ft 3 . What is the width of the tank to the nearest tenth of a foot? Chapter 9 268 ℓ ft 4 ft w Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 15. Complete the reasoning model below. Think Write I need to use the equation for the volume V= of a rectangular prism. I will substitute 500 for V, 2w for 𝓵𝓵, and 500 = (2w) ? 4 for h. 500 = I need to simplify. Finally, I need to isolate w and find ± the square roots of each side. I will use a calculator. √ 8 ± 16. The width of the tank is about ? ? w2 =w ≈w ft. Lesson Check HSM11_A1MC_0903_T91321 • Do you UNDERSTAND? Compare and Contrast When is it easier to solve a quadratic equation of the form ax2 1 c 5 0 using square roots than to solve it using a graph? 17. Solve each equation below using square roots. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 5x 2 2 50 5 0 5x 2 2 125 5 0 5x 2 5 5x 2 5 x 2 5 x 2 5 x 5 4 x 5 4 18. The equation from Exercise 17 that is easy to solve using a graph is . 19. When is it easier to solve an equation of the form ax 2 1 c 5 0 using square roots? ___________________________________________________________________________________ Math Success Check off the vocabulary words that you understand. standard form of a quadratic equation root of an equation zero of a function Rate how well you can solve a quadratic equation. Need to review 0 2 4 6 8 Now I get it! 10 269 Lesson 9-3