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5.3 Solving Quadratic Equations by Finding Square Roots

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EXAMPLE 1 Use properties of square roots Simplify the expression. 5= 4a. 80 5 16 = = 3 14b. 6 21 126 = 9 14 = c. 4 81 = 4 = 2929 d. 7 16 = 7 = 4 16 7

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1.2. 1._____________ 2._____________ 3.4. 1.____________ 2.____________ 5.6. 1._____________ 2._____________ 7.8. 1._____________ 2._____________ 9.10. 9._____________ 10._____________ 11.12. 9._____________ 10._____________ Show all work! Simplify each of the following. Leave each answer in radical form.

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GUIDED PRACTICE for Example 1 27 1. 3 3 98 2. 2 7 ANSWER 3. 6 510 15 4. 8 2814 4 ANSWER

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GUIDED PRACTICE for Example 1 5. 3 8 9 64 6. 15 4 2 ANSWER 7. 11 25 8. 36 49 11 5 7 6 ANSWER

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EXAMPLE 2 Rationalize denominators of fractions. Simplify (a) 5 2 and 3 7 + 2 SOLUTION (a) 5 2 2 10 = = 5 2 = 5 2 2 2 (b) SOLUTION (b) 3 7 + 2 = 3 2 7 – 2 2 = 21 – 3 2 49 – 7 + 722 - 2 = 21 – 3 2 47

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EXAMPLE 3 Solve a quadratic equation Solve 3x 2 + 5 = 41. 3x 2 + 5 = 41 Write original equation. 3x 2 = 36 Subtract 5 from each side. x 2 = 12 Divide each side by 3. x = + 12 Take square roots of each side. x = + 43 + 2 3 Product property Simplify. - 5 -5 3

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EXAMPLE 3 Solve a quadratic equation ANSWER The solutions are and 2 3 2 3 – Check the solutions by substituting them into the original equation. 3x 2 + 5 = 41 3( ) 2 + 5 = 41 23 ? 41 = 41 3(12) + 5 = 41 ? 3x 2 + 5 = 41 3( ) 2 + 5 = 41 – 23 ? 41 = 41 3(12) + 5 = 41 ?

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EXAMPLE 4 Standardized Test Practice SOLUTION 1515 (z + 3) 2 = 7 Write original equation. (z + 3) 2 = 35 Multiply each side by 5. z + 3 = + 35 Take square roots of each side. z = –3 + 35 Subtract 3 from each side. The solutions are –3 + and –3 – 35 ANSWER The correct answer is C.

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GUIDED PRACTICE for Examples 2, 3, and 4 GUIDED PRACTICE Simplify the expression. 9. 5 30 6 5 10. 9 8 2 4 3 11. 17 12 51 6 ANSWER 12. 399 21 19 21 – 21 – 3 5 22 13. – 6 7 – 5 8 – 2 11 5 14. 2 4 + 11 ANSWER

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– 9 +7 74 for Examples 2, 3, and 4 GUIDED PRACTICE 15. – 1 9 + 7 32 + 4 3 61 16. 4 8 – 3 ANSWER

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for Examples 2, 3, and 4 GUIDED PRACTICE 17. Solve the equation. 5x 2 = 80 + 4 18. z 2 – 7 = 29 + 6 19. 3(x – 2) 2 = 40 120 3 2 + ANSWER

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EXAMPLE 5 Model a dropped object with a quadratic function Science Competition For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground ? SOLUTION h = – 16t 2 + h 0 Write height function. 0 = – 16t 2 + 50 Substitute 0 for h and 50 for h 0. – 50 = – 16t 2 Subtract 50 from each side. 50 16 = t 2 Divide each side by –16. 50 16 + = t Take square roots of each side. + 1.8 t Use a calculator. ANSWER Reject the negative solution, –1.8, because time must be positive. The container will fall for about 1.8 seconds before it hits the ground.

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Homework: Simplifying radicals worksheet and Pg. 267 #47-67o, 69, 70

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