1. Diamond Factoring

2. When we factor a trinomial we are undoing FOIL With FOIL (x + 2) (x – 5) = x 2 – 3x – 10 Undoing FOIL X 2 – 3x – 10 = (x + 2) (x – 5) Where do the 2 and -5 come from? They are factors of -10 whose sum is -3

3. Diamond 3-9 Product Sum

4. Warm-Up Please complete these individually. 1. Fill in the following diamonds. In a diamond the top number is the product of the a. b. 15 c. -8 2.Write the general form of a quadratic equation. ax 2 + bx + c = 0

5. Diamond Factoring This is a guaranteed method for factoring quadratic equations—no guessing necessary! We will learn how to factor quadratic equations using the diamond method Background knowledge needed: –Basic x-solve problems –General form of a quadratic equation

6. Factor the diamond way Example: Factor x 2 -3x -10 -3 -10 -5 2 x 2 -3x -10 = (x-5)(x+2)

7. Factor the diamond way Middle b=m+n Sum Product c=mn m n First and Last Coefficients y = x 2 + bx + c Y = (x + m) (x + n)

8. Examples Factor using the diamond method. 1. x 2 + 4x – 12 a) -12 4 6 -2 Solution: x 2 + 4x – 12 = (x + 6)(x - 2)

9. Examples continued 2. x 2 - 9x + 20 a) 20 -9 Solution: x 2 - 9x + 20 = (x - 4)(x - 5 ) -4 -5

10. Think-Pair-Share 1.Based on the problems we’ve done, list the steps in the diamond factoring method so that someone else can do a problem using only your steps. 2. Trade papers with your partner and use their steps to factor the following problem: x 2 +4x -32.

11. Trying out the Steps 3.If you cannot complete the problem using only the steps written, put an arrow on the step where you stopped. Give your partner’s paper back to him. 4.Modify the steps you wrote to correct any incomplete or incorrect steps. Finish the problem based on your new steps and give the steps back to your partner. 5.Try using the steps again to factor: x 2 +4x +3.

12. Stepping Up 6.Try out your steps and factor: x 2 + 1x – 20.

13. Examples continued 3. x 2 - 6x - 7 a) -7 -6 Solution: x 2 - 6x – 7 = (x - 7)(x + 1 ) -7 1

14. Examples continued 3. x 2 - 7x + 12 a) 12 -7 Solution: x 2 - 7x + 12 = (x - 4)(x - 3 ) -4 -3

15. Now use this to solve an equation 1.Make sure all terms of the polynomial are on the left side. 2.Divide all terms by a negative 1 if necessary to make the lead coefficient positive. 3.Factor using the diamond method. 4.Set the factors equal to zero and solve.

16. Solving an equation with a trinomial Solve x 2 + 4x + 3 = 0 Factor (x + 3)(x + 1) = 0 X + 3 = 0 x + 1 = 0 x = -3 x = -1 {-3, -1} (note: not ( ) )