Download Diamond and Box Factoring

Solving Quadratic
Equations by
Factoring
X- Box
Product
a∙c
factors
factors
Sum
b
X-box Factoring
• This is a guaranteed method for
factoring quadratic equations—no
guessing necessary!
• We will learn how to factor quadratic
equations using the x-box method
• Background knowledge needed:
– Basic x-solve problems
– General form of a quadratic equation
ax  bx  c  0
2
Solve the x-box way
Example: Factor 3x2 -13x -10 = 0
x
-5
3x
3x2
-15x
+2
2x
-10
(3)(-10)=
-30
2
-15
-13
3x2 -13x -10 = 0
(x-5)(3x+2) = 0
X = 5, -2/3
FACTOR the x-box way
ax2 + bx + c
First and
Last
Coefficients
Product
GCF
GCF
ac=mn
GCF
1st
Term
Factor
n
GCF
Factor
m
Last
term
n
m
b=m+n
Sum
Middle
Examples
Solve using the x-box method.
1. x2 + 4x – 12 = 0
6
-12
4
-2
x
+6
x
x2
6x
-2
-2x
-12
Solution: x2 + 4x – 12 = 0
(x + 6)(x - 2) = 0
x = -6,2
Examples
2. x2 - 9x + 20 = 0
20
-4 -5
-9
x
-4
x
x²
-4x
-5
-5x
20
Solution: x2 - 9x + 20 = 0
(x - 4)(x - 5) = 0
x = 4, 5
Examples
3. 2x2 - 5x – 7 = 0
-14
-7 2
-5
2x
x
+1
2x²
2x
Solution: 2x2 - 5x – 7 = 0
(2x - 7)(x + 1) = 0
x = 7/2, -1
-7
-7x
-7
Examples
4. 15x2 + 7x – 2 = 0
-30
10 -3
7
5x
-1
3x
+2
15x²
10x
-3x
-2
Solution: 15x2 + 7x – 2 = 0
(3x + 2)(5x - 1) = 0
x = -2/3, 1/5