1. Lesson 2-3 The Quadratic Equation Objective: To learn the various ways to solve quadratic equations, including factoring, completing the square and the quadratic formula.

2. Quadratic Equations  A quadratic equation in x is an equation that can be written in the general form ax 2 + bx + c = 0  where a, b, and c are real numbers  We can solve by several methods: By Factoring and setting each factor equal to 0 Extracting Square Roots Completing the Square Using the Quadratic Formula

3. Solving by Factoring  Factor ax 2 + bx + c =(Ax+B)(Cx+D) = 0 Set each factor = 0. (Ax+B) = 0, (Cx+D) = 0 Solve for x

4. Solving by Factoring xx 2 -7x + 12=0 find factors of 12 that add to -7  (x – 3)(x – 4) = 0  x-3 = 0 x–4 =0 = 3 x = 4

5. Solving by Factoring  2x 2 – 7x -15 = 0 (2x + 3)( x- 5) = 0 2x + 3 = 0 x – 5 = 0 2x = -3 x = x = 5

6. Solving by Factoring  4x 2 – 3x = 0 x(4x – 3) = 0 x = 0 4x -3 = 0 4x = 3 x =

7. Practice  4x 2 – x = 0  3x 2 – 11x -4 = 0

8. Solving by Completing the Square Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation

9. Solving by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

10. Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation. Solving by Completing the Square

11. Step 4: Take the square root of each side

12. Solving by Completing the Square Step 5: Set up the two possibilities and solve

13. Completing the Square-Example #2 Solve the following equation by completing the square: Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.

14. Solving by Completing the Square Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides. The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.

15. Solving Completing the Square Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.

16. Solving by Completing the Square Step 4: Take the square root of each side

17. Solving by Completing the Square Try the following examples. Do your work on your paper and then check your answers.

18. Warm up  Solve by factoring:  x 2 + 5x +6=0  2x 2 + 9x – 18 = 0  3x 2 +x =0

19. Taking Square Roots  x 2 – 3 = 0 (can’t factor) x 2 = 3 take the square root of both sides x = √3 or x = - √3 x = ±√3

20. Taking Square Roots  x 2 + 9 = 0 x 2 = -9 x = √-9 x = ±3 i

21. Taking Square Roots  2(x – 1) 2 – 4 = 0 +4 +4 2(x – 1) 2 = 4 2 2 (x – 1) 2 = 2 take the sq. root of each side x – 1 = ±√2 +1 +1 x = 1±√2

22. Practice  5x 2 + 13 = 0  (2x – 7) 2 – 5 = 0

23. The Quadratic Formula Song x equals negative b plus or minus the square root of b squared minus 4ac all over 2a

24. Quadratic Formula  Let’s look at an example.  3x 2 - 4x + 3 = 0  a = ?  b = ?  c = ?  a = 3  b = -4  c = 3

25. Quadratic Formula  Now let’s plug it in.  b = -4, so -b = -(-4) = 4

26. Quadratic Formula Simplify

27. Quadratic Formula Find the zeros of r 2 - 7r -18 = 0

28. Quadratic Formula Simplify

29. Quadratic Formula Now let’s examine our solution. We can break this into two equations.

30. Quadratic Formula Now we can get our two solutions.

31. Quadratic Formula  x 2 – 8x = -10  4x 2 -2x +1 = 0