1. Solving Quadratic Equations by Graphing Chapter 9.2

2. Solutions to Quadratic Equations

3. Two Roots Example Solve x 2 – 3x – 10 = 0 by graphing. Graph the function (using “the steps”). f(x) = x 2 – 3x – 10. The x-intercepts of the parabola appear to be at –2 and 5. So the solutions are –2 and 5.

4. Two Roots – Your Turn! A.{–2, 4} B.{2, –4} C.{2, 4} D.{–2, –4} Solve x 2 – 2x – 8 = 0 by graphing.

5. Double Root Example Solve x 2 + 8x = –16 by graphing. Step 1First, rewrite the equation so one side is equal to zero. x 2 + 8x=–16Original equation x 2 + 8x + 16=–16 + 16Add 16 to each side. x 2 + 8x + 16=0Simplify.

6. Double Root Example (Cont.) Step 2Graph the related function f(x) = x 2 + 8x + 16.

7. Double Root Example (Cont. 2) Step 3Locate the x-intercepts of the graph. Notice that the vertex of the parabola is the only x-intercept. Therefore, there is only one solution, –4. Answer: The solution is –4. CheckSolve by factoring. x 2 + 8x + 16=0Original equation (x + 4)(x + 4)=0Factor. x + 4 = 0 or x + 4 = 0Zero Product Property x = –4 x = –4Subtract 4 from each side.

8. Double Root – Your Turn! Solve x 2 + 2x = –1 by graphing. A.{1} B.{–1} C.{–1, 1} D.Ø

9. No Real Roots Example Solve x 2 + 2x + 3 = 0 by graphing. Graph the related function f(x) = x 2 + 2x + 3. The graph has no x-intercept. Thus, there are no real number solutions for the equation. Answer: The solution set is {Ø}.

10. Your Turn! Solve x 2 + 4x + 5 = 0 by graphing. A.{1, 5} B.{–1, 5} C.{5} D.Ø

11. Solve x 2 – 4x + 2 = 0 by graphing. If integral roots cannot be found, estimate the roots. Graph the related function f(x) = x 2 – 4x + 2. Approximate the Roots What numbers are the answers between? Between 0 and 1 Between 3 and 4

12. Completing the Square Chapter 9.4

13. Completing the Square

14. Complete the Square x 2 – 12x + c Method Complete the square. Answer: Thus, c = 36. Notice that x 2 – 12x + 36 = (x – 6) 2. Step 1 Step 2Square the result(–6) 2 = 36 of Step 1. Step 3Add the result ofx 2 –12x + 36 Step 2 to x 2 – 12x.

15. Complete the Square – Your Turn! A.7 B.14 C.156 D.49 Find the value of c that makes x 2 + 14x + c a perfect square.

16. Solve by Completing the Square Solve x 2 + 6x + 5 = 12 by completing the square. Isolate the x 2 - and x-terms. Then complete the square and solve. x 2 + 6x + 5 = 12Original equation x 2 + 6x – 5 – 5= 12 – 5 Subtract 5 from each side. x 2 + 6x = 7Simplify. x 2 + 6x + 9 =7 + 9

17. Solve by Completing the Square (Cont.) (x + 3) 2 =16Factor x 2 + 6x + 9. = –7 = 1 Simplify. Answer: The solutions are –7 and 1. x + 3 =±4 Take the square root of each side. x + 3 – 3=±4 – 3Subtract 3 from each side. x=±4 – 3 Simplify. x = –4 – 3 or x = 4 – 3Separate the solutions.

18. Solve by Completing the Square – Your Turn! A.{–2, 10} B.{2, –10} C.{2, 10} D.Ø Solve x 2 – 8x + 10 = 30.

19. A.{–1} B.{–1, –7} C.{–1, 7} D.Ø Solve x 2 + 8x + 10 = 3 by completing the square. Complete the Square – Your Turn!

20. Complete the Square (a = 0) Solve –2x 2 + 36x – 10 = 24 by completing the square. –2x 2 + 36x – 10 = 24Original equation Isolate the x 2 - and x-terms. Then complete the square and solve. x 2 – 18x + 5= –12Simplify. x 2 – 18x + 5 – 5= –12 – 5 Subtract 5 from each side. x 2 – 18x = –17Simplify. Divide each side by –2.

21. (x – 9) 2 =64Factor x 2 – 18x + 81. = 17 = 1Simplify… Solutions! x – 9 =±8 Take the square root of each side. x – 9 + 9 =±8 + 9Add 9 to each side. x =9 ± 8 Simplify. x = 9 + 8 or x = 9 – 8Separate the solutions. x 2 – 18x + 81 =–17 + 81 Complete the Square (a = 0) (Cont.)