1. 9-8 The Quadratic Formula Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 9-8 The Quadratic Formula Warm Up Evaluate for x = –2, y = 3, and z = –1. 6 1. x 2 2. xyz 3. x 2 – yz4. y – xz 4 5. –x6. z 2 – xy 71 7 2

3. 9-8 The Quadratic Formula California Standards 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 20.0 Students use the quadratic formula to find the roots of a second- degree polynomial and to solve quadratic equations.

4. 9-8 The Quadratic Formula In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax 2 + bx + c = 0, you can derive the Quadratic Formula.

5. 9-8 The Quadratic Formula

6. 9-8 The Quadratic Formula

7. 9-8 The Quadratic Formula

8. 9-8 The Quadratic Formula To add fractions, you need a common denominator. Remember!

9. 9-8 The Quadratic Formula Additional Example 1A: Using the Quadratic Formula Solve using the Quadratic Formula. 6x 2 + 5x – 4 = 0 6x 2 + 5x + (–4) = 0 Identify a, b, and c. Use the Quadratic Formula. Simplify. Substitute 6 for a, 5 for b, and –4 for c.

10. 9-8 The Quadratic Formula Additional Example 1A Continued Solve using the Quadratic Formula. 6x 2 + 5x – 4 = 0 Simplify. Write as two equations. Solve each equation.

11. 9-8 The Quadratic Formula Additional Example 1B: Using the Quadratic Formula Solve using the Quadratic Formula. x 2 = x + 20 1x 2 + (–1x) + (–20) = 0 Write in standard form. Identify a, b, and c. Use the Quadratic Formula. Simplify. Substitute 1 for a, –1 for b, and –20 for c.

12. 9-8 The Quadratic Formula Additional Example 1B Continued Solve using the Quadratic Formula. x = 5 or x = –4 Simplify. Write as two equations. Solve each equation. x 2 = x + 20

13. 9-8 The Quadratic Formula You can graph the related quadratic function to see if your solutions are reasonable. Helpful Hint

14. 9-8 The Quadratic Formula Check It Out! Example 1a Solve using the Quadratic Formula. Check your answer. –3x 2 + 5x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute –3 for a, 5 for b, and 2 for c. Simplify. –3x 2 + 5x + 2 = 0

15. 9-8 The Quadratic Formula Check It Out! Example 1a Continued Solve using the Quadratic Formula. Check your answer. Simplify. Write as two equations. Solve each equation. x = – or x = 2 –3x 2 + 5x + 2 = 0

16. 9-8 The Quadratic Formula Check It Out! Example 1a Continued Solve using the Quadratic Formula. Check your answer. Check –3x 2 + 5x + 2 = 0 x = – or x = 2

17. 9-8 The Quadratic Formula Check It Out! Example 1b Solve using the Quadratic Formula. Check your answer. 2 – 5x 2 = –9x Write in standard form. Identify a, b, and c. (–5)x 2 + 9x + (2) = 0 Use the Quadratic Formula. Substitute –5 for a, 9 for b, and 2 for c. Simplify

18. 9-8 The Quadratic Formula Check It Out! Example 1b Continued Solve using the Quadratic Formula. Check your answer. Simplify. Write as two equations. Solve each equation. 2 – 5x 2 = –9x x = – or x = 2

19. 9-8 The Quadratic Formula Check It Out! Example 1b Continued Solve using the Quadratic Formula. Check your answer. –5(2) 2 + 9(2) + 2 0 –20 + 18 + 2 0 0 Check –5x 2 + 9x + 2 = 0 –5 + 9 + 2 0 0 + 2 0

20. 9-8 The Quadratic Formula Because the Quadratic Formula contains a square root, the solutions may be irrational. You can give the exact solution by leaving the square root in your answer, or you can approximate the solutions.

21. 9-8 The Quadratic Formula Additional Example 2: Using the Quadratic Formula to Estimate Solutions Solve x 2 + 3x – 7 = 0 using the Quadratic Formula. Check reasonableness Estimate : x ≈ 1.54 or x ≈ –4.54.

22. 9-8 The Quadratic Formula Check It Out! Example 2 Solve 2x 2 – 8x + 1 = 0 using the Quadratic Formula. Check reasonableness Estimate : x ≈ 3.87 or x ≈ 0.13.

23. 9-8 The Quadratic Formula There is no one correct way to solve a quadratic equation. Many quadratic equations can be solved using several different methods: graphing, factoring, completing the square, using square roots, and using the Quadratic Formula.

24. 9-8 The Quadratic Formula Additional Example 3: Solving Using Different Methods Solve x 2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 1 Solve by graphing. y = x 2 – 9x + 20 Write the related quadratic function and graph it. The solutions are the x- intercepts, 4 and 5.

25. 9-8 The Quadratic Formula Additional Example 3 Continued Solve x 2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 2 Solve by factoring. x 2 – 9x + 20 = 0 (x – 4)(x – 5) = 0 x – 4 = 0 or x – 5 = 0 x = 4 or x = 5 Factor. Use the Zero Product Property. Solve each equation.

26. 9-8 The Quadratic Formula Additional Example 3 Continued Solve x 2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Check: 4 and 5 Check x 2 – 9x + 20 = 0 (4) 2 – 9(4) + 20 0 16 – 36 + 20 0 0 (5) 2 – 9(5) + 20 0 25 – 45 + 20 0 0

27. 9-8 The Quadratic Formula Check It Out! Example 3a Solve. Show your work and check your answer. x 2 + 7x + 10 = 0 Method 3 Solve by completing the square. x 2 + 7x + 10 = 0 x 2 + 7x = –10 Add to both sides. Factor and simplify. Take the square root of both sides.

28. 9-8 The Quadratic Formula Check It Out! Example 3a Continued Solve. Show your work and check your answer. x 2 + 7x + 10 = 0 Solve each equation. or x = –2 or x = –5 (–2) 2 + 7(–2) + 10 0 4 – 14 + 10 0 0 (–5) 2 + 7(–5) + 10 0 25 – 35 + 10 0 0 Check x 2 + 7x + 10 = 0 x 2 + 7x + 10 = 0

29. 9-8 The Quadratic Formula Check It Out! Example 3b Solve. Show your work and check your answer. –14 + x 2 – 5x = 0 Method 4 Solve using the Quadratic Formula. x 2 – 5x – 14 = 0 1x 2 – 5x – 14 = 0 Identify a, b, and c. Substitute 1 for a, –5 for b, and –14 for c. Simplify.

30. 9-8 The Quadratic Formula Check It Out! Example 3b Continued Solve. Show your work and check your answer. –14 + x 2 – 5x = 0 x = 7 or x = –2 or Write as two equations. Solve each equation.

31. 9-8 The Quadratic Formula Check It Out! Example 3b Continued Solve. Show your work and check your answer. –14 + x 2 – 5x = 0 x 2 – 5x – 14 = 0 7 2 – 5(7) – 14 0 49 – 35 – 14 0 14 – 14 0 0 Check x 2 – 5x – 14 = 0 –2 2 – 5(–2) – 14 0 4 + 10 – 14 0 14 – 14 0 0

32. 9-8 The Quadratic Formula Check It Out! Example 3c Solve. Show your work and check your answer. 2x 2 + 4x – 21 = 0 Method 1 Solve by graphing. 2x 2 + 4x – 21 = y Write the related quadratic function. Divide each term by 2 and graph. The solutions are the x-intercepts and appear to be ≈ 2.4 and ≈ –4.4.

33. 9-8 The Quadratic Formula Check It Out! Example 3c Continued Solve. Show your work and check your answer. 2x 2 + 4x – 21 = 0 Check y = 2x 2 + 4x – 21 using a calculator. Answers x ≈ –4.39 and ≈ 2.39 appear to be reasonable.

34. 9-8 The Quadratic Formula Notice that all of the methods in Example 3 on p. 600 produce the same solution, –1 and –6. The only method you cannot use to solve x 2 + 7x + 6 = 0 is using square roots. Sometimes one method is better for solving certain types of equations. The table below gives some advantages and disadvantages of the different methods.

35. 9-8 The Quadratic Formula

36. 9-8 The Quadratic Formula 1. Solve x 2 + x = 12 by using the Quadratic Formula. 2. Solve –3x 2 + 5x = 1 by using the Quadratic Formula. 3. Solve 8x 2 – 13x – 6 = 0. Use at least 2 different methods. Lesson Quiz 3, –4 = 0.23, ≈ 1.43