Algebra 1B Chapter 9

1. Algebra 1BChapter 9 Solving Quadratic Equations The Quadratic Formula

2. Warm Up Evaluate for x = –2, y = 3, and z = –1. 1. x2 4 2. xyz 6 3. x2 – yz 4. y – xz 7 1 5. –x 6. z2 – xy 7 2

3. In the previous lesson, you completed the square to solve quadratic equations. If you complete the square of ax2 + bx + c = 0, you can derive the Quadratic Formula.

5. Remember! To add fractions, you need a common denominator.

6. Additional Example 1A: Using the Quadratic Formula Solve using the Quadratic Formula. 6x2 + 5x – 4 = 0 6x2 + 5x + (–4) = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute 6 for a, 5 for b, and –4 for c. Simplify.

7. Additional Example 1A Continued Solve using the Quadratic Formula. 6x2 + 5x – 4 = 0 Simplify. Write as two equations. Solve each equation.

8. Additional Example 1B: Using the Quadratic Formula Solve using the Quadratic Formula. x2 = x + 20 Write in standard form. Identify a, b, and c. 1x2 + (–1x) + (–20) = 0 Use the Quadratic Formula. Substitute 1 for a, –1 for b, and –20 for c. Simplify.

9. Additional Example 1B Continued Solve using the Quadratic Formula. x2 = x + 20 Simplify. Write as two equations. Solve each equation. x = 5 or x = –4

10. In Your Notes! Example 1a Solve using the Quadratic Formula. Check your answer. –3x2 + 5x + 2 = 0 –3x2 + 5x + 2 = 0 Identify a, b, and c. Use the Quadratic Formula. Substitute –3 for a, 5 for b, and 2 for c. Simplify.

11. x = – or x = 2 In Your Notes! Example 1a Continued Solve using the Quadratic Formula. Check your answer. –3x2 + 5x + 2 = 0 Simplify. Write as two equations. Solve each equation.

12. In Your Notes! Example 1b Solve using the Quadratic Formula. Check your answer. 2 – 5x2 = –9x (–5)x2 + 9x + (2) = 0 Write in standard form. Identify a, b, and c. Use the Quadratic Formula. Substitute –5 for a, 9 for b, and 2 for c. Simplify

13. or x = 2 x = – In Your Notes! Example 1b Continued Solve using the Quadratic Formula. Check your answer. 2 – 5x2 = –9x Simplify. Write as two equations. Solve each equation.

14. –5x2 + 9x + 2 = 0 Check –5x2 + 9x + 2 = 0 –5(2)2 + 9(2) + 2 0 –5 + 9 + 2 0 –20 + 18 + 2 0 0 0 + 2 0  0 0 In Your Notes! Example 1b Continued Solve using the Quadratic Formula. Check your answer. 

15. 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.

16. Estimate : x ≈ 1.54 or x ≈ –4.54. Additional Example 2: Using the Quadratic Formula to Estimate Solutions Solve x2 + 3x – 7 = 0 using the Quadratic Formula. Check reasonableness

17. Estimate : x ≈ 3.87 or x ≈ 0.13. In Your Notes! Example 2 Solve 2x2 – 8x + 1 = 0 using the Quadratic Formula. Check reasonableness

18. 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 • Square roots and using • the Quadratic Formula

19. Additional Example 3: Solving Using Different Methods Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 1 Solve by graphing. Write the related quadratic function and graph it. y = x2 – 9x + 20 The solutions are the x-intercepts, 4 and 5.

20. Additional Example 3 Continued Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Method 2 Solve by factoring. x2 – 9x + 20 = 0 (x – 4)(x – 5) = 0 Factor. x – 4 = 0 or x – 5 = 0 Use the Zero Product Property. x = 4 or x = 5 Solve each equation.

21. x2 – 9x + 20 = 0 Check x2 – 9x + 20 = 0 (5)2 – 9(5) + 20 0 (4)2 – 9(4) + 20 0 25 – 45 + 20 0 16 – 36 + 20 0   0 0 0 0 Additional Example 3 Continued Solve x2 – 9x + 20 = 0. Show your work. Use at least two different methods. Check your answer. Check: 4 and 5

22. Add to both sides. x2 +7x = –10 In Your Notes! Example 3a Solve. Show your work and check your answer. x2 + 7x + 10 = 0 Method 3 Solve by completing the square. x2 + 7x + 10 = 0 x2 + 7x = –10 Factor and simplify. Take the square root of both sides.

23. or x2 + 7x + 10 = 0 Check x2 + 7x + 10 = 0 (–2)2 + 7(–2) + 10 0 (–5)2 + 7(–5) + 10 0 4 – 14 + 10 0 25 – 35 + 10 0   0 0 0 0 In Your Notes! Example 3a Continued Solve. Show your work and check your answer. x2 + 7x + 10 = 0 Solve each equation. x = –2 or x = –5

24. In Your Notes! Example 3b Solve. Show your work and check your answer. –14 + x2 – 5x = 0 Method 4 Solve using the Quadratic Formula. x2 – 5x – 14 = 0 1x2–5x –14 = 0 Identify a, b, and c. Substitute 1 for a, –5 for b, and –14 for c. Simplify.

25. or In Your Notes! Example 3b Continued Solve. Show your work and check your answer. –14 + x2 – 5x = 0 Write as two equations. x = 7or x = –2 Solve each equation.

26. x2 – 5x – 14 = 0 Check 72 – 5(7) – 14 0 –22 – 5(–2) – 14 0 49 – 35 – 14 0 4 + 10 – 14 0 14 – 14 0 14 – 14 0 0 0 0 0 In Your Notes! Example 3b Continued Solve. Show your work and check your answer. –14 + x2 – 5x = 0 x2 – 5x – 14 = 0  

27. In Your Notes! Example 3c Solve. Show your work and check your answer. 2x2 + 4x – 21 = 0 Method 1 Solve by graphing. 2x2 + 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.

28. Sometimes one method is better for solving certain types of equations. The table below gives some advantages and disadvantages of the different methods.