Diamond Factoring When we factor a trinomial we are undoing FOIL With FOIL (x + 2) (x 5) = x2 3x 10 Undoing FOIL X2 3x 10 = (x + 2) (x 5) Where do the 2 and -5 come from? They are factors of -10 whose sum is -3
Diamond Product 3 -9 Sum 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. 4 15 7 -8
c. 36 13 Write the general form of a quadratic equation. ax2 + bx + c = 0 Diamond Factoring This is a guaranteed method for factoring quadratic equationsno
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 Factor the diamond way Example: Factor x2 -3x -10 -10 2
-5 -3 x2 -3x -10 = (x-5)(x+2) Factor the diamond way y = x2 + bx + c First and Last Coefficients
Produ ct m c=mn n Middle b=m+ n Sum
Y = (x + m) (x + n) Examples Factor using the diamond method. 1. x2 + 4x 12 a) -12 6 -2
-4 -5 -9 Solution: x2 - 9x + 20 = (x - 4)(x - 5) Think-Pair-Share 1. Based on the problems weve 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: x2 +4x -32. 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 partners 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:
x2 +4x +3. Stepping Up 6. Try out your steps and factor: x2 + 1x 20. Examples continued continued 3. x2 - 6x - 7 a) -7
-7 1 -6 Solution: x2 - 6x 7 = (x - 7)(x + 1) Examples continued continued 3. x2 - 7x + 12 a) 12
-4 -3 -7 Solution: x2 - 7x + 12 = (x - 4)(x - 3) 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. Solving an equation with a trinomial Solve x2 + 4x + 3 = 0 Factor (x + 3)(x + 1) = 0 X+3=0 x+1=0 x = -3 x = -1
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