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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
2.
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 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
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
Product
c=mn
n
m
b=m+n
Sum
Middle
Y = (x + m) (x + n)
Examples
Factor using the diamond method.
1. x2 + 4x – 12
a)
6
-12
4
-2
Solution: x2 + 4x – 12 = (x + 6)(x - 2)
Examples
continued
2. x2 - 9x + 20
a)
20
-4
-5
-9
Solution: x2 - 9x + 20 = (x - 4)(x - 5)
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: 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
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:
x2 +4x +3.
Stepping Up
6. Try out your steps and factor:
x2 + 1x – 20.
Examples continued
3. x2 - 6x - 7
a)
-7
-7
-6
1
Solution: x2 - 6x – 7 = (x - 7)(x + 1)
Examples continued
3. x2 - 7x + 12
a)
-4
12
-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
{-3, -1} (note: not ( ) )