1. 10.6 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Solve Quadratic Equations by the Quadratic Formula

2. 10.6 Warm-Up Evaluate the expression for the given value of x. 1.15 – (–x) + 9; x = –2 2.14 – x + 3; x = 8 ANSWER 22 ANSWER 9 22 ft A basketball is thrown in the air from a height of 6 feet with an initial velocity of 40 feet per second. What is the height (in feet) of the ball after 2 seconds? 3.

3. 10.6 Example 1 What are the solutions of 3x 2 + 5x = 8? –1 and – A 8 3 B –1 and 8 3 C 1 and – 8 3 D 1 and 8 3 SOLUTION Write original equation. 3x 2 + 5x = 8 Write in standard form. 3x 2 + 5x – 8 = 0 Quadratic formula ± x = –bb 2 – 4ac 2a2a

4. 10.6 Example 1 Substitute values in the quadratic formula : a = 3, b = 5, and c = – 8. Simplify. = –5 ± 121 6 Simplify the square root. = –5 ± 11 6 The solutions of the equation are –5 + 11 6 = 1 and –5 – 11 6 = – 8 3 x = –5 ± 5 2 – 4(3)(–8) 2(3) The correct answer is C. ANSWER

5. 10.6 Example 2 2x 2 – 7 = x 2x 2 – x – 7 = 0 Solve 2x 2 – 7 = x. Write original equation. Write in standard form. Quadratic formula x =x = b 2 – 4ac + – –b 2a2a Substitute values in the quadratic formula: a = 2, b = –1, and c = –7. Simplify. – (–1) – + ( –1) 2 – 4(2)(–7) 2(2) = 4 = + – 1 57

6. 10.6 Example 2 ANSWER CHECK 1 + 57 2.14 and 1 – 4 –1.64. 4 The solutions are 57 Write the equation in standard form, 2x 2 – x – 7 = 0. Then graph the related function y = 2x 2 – x – 7. The x- intercepts are about –1.6 and 2.1. So, each solution checks.

7. 10.6 Guided Practice Use the quadratic formula to solve the equation. Round your solution to the nearest hundredth, if necessary. 1. x 2 – 8x + 16 = 0 ANSWER 4 2. 3n 2 – 5n = –1 ANSWER 0.23, 1.43 3. 4z 2 = 7z + 2 ANSWER  0.25, 2

8. 10.6 Example 3 y = 10x 2 – 94x + 3900 4200 = 10x 2 – 94x – 3900 0 = 10x 2 – 94x – 300 Write function. Substitute 4200 for y. Write in standard form. SOLUTION For the period 1971  2001, the number y of films produced in the world can be modeled by the function y = 10x 2  94x + 3900 where x is the number of years since 1971. In what year were 4200 films produced ? FILM PRODUCTION

9. 10.6 Example 3 Substitute values in the quadratic formula: a = 10, b = –94, and c = –300. 20,836 94 + – 20 = Simplify. ANSWER There were 4200 films produced about 12 years after 1971, or in 1983. x = (–94) 2 – 4 (10)(–300) –(–94) + – 2(10) The solutions of the equation are: 94 + 20,836 20 12 94 –20,836 20 –3. and

10. 10.6 Guided Practice WHAT IF? In Example 3, find the year when 4750 films were produced. 4. ANSWER 1986

11. 10.6 Example 4 Tell what method you would use to solve the quadratic equation. Explain your choice(s). a. 10x 2 – 7 = 0 SOLUTION a. The quadratic equation can be solved using square roots because the equation can be written in the form x 2 = d. b. x 2 + 4x = 0 b. The equation can be solved by factoring because the expression x 2 + 4x can be factored easily. Also, the equation can be solved by completing the square because the equation is of the form ax 2 + bx + c = 0 where a = 1 and b is an even number. c. 5x 2 + 9x – 4 = 0

12. 10.6 Example 4 c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation can be solved using the quadratic formula. SOLUTION

13. 10.6 Guided Practice Tell what method you would use to solve the quadratic equation. Explain your choice(s). 5. x 2 + x – 6 = 0 ANSWER Factoring because the expression factors easily. 6. x 2 – 9 = 0 ANSWER Factoring because the expression factors easily. Using square roots is another option since the equation can be written in the form x 2 = d.

14. 10.6 Guided Practice 7. x 2 + 6x = 5 ANSWER Completing the square because the equation is of the form ax 2 + bx = c where a = 1 and b is an even number. Another method is the quadratic formula since the equation does not factor easily. Tell what method you would use to solve the quadratic equation. Explain your choice(s).

15. 10.6 Lesson Quiz 1. 6x 2 – 6 = –5x Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary. 2. 2x 2 + 3x – 8 = 0 –2.87, 1.39 ANSWER, 3 2 – 2 3 3. 3m 2 – m + 9 = 12 ANSWER –0.85, 1.18

16. 10.6 Lesson Quiz You have seen that the function y = 10x 2  94x + 3900 models the number y of films produced in the world, where x is the number of years since 1971. In what year were 10,000 films produced ? 4. ANSWER about 29.8 years after 1971, or during the year 2000