Middle School Math Solutions – Polynomials Calculator, Factoring Quadratics

Just like numbers have factors (2×3=6), expressions have factors ((x+2)(x+3)=x^2+5x+6). Factoring is the process of finding factors of a number or expression, where we find what multiplies together to make the expression or number.  In today’s blog post, we will talk about how to factor simple expressions and quadratics.

Factoring simple expressions – 
Given a simple expression, ax+b, pull out the greatest common factor from the expression. Pretty simple!
Here’s an example (click here):

Factor\:2x+6
2x+2\cdot3
2(x+3)

Factoring quadratics – 
Quadratics have the form: (ax)^2+bx+c, where a, b, and c are numbers.
Here are the steps for factoring quadratics:

1. Find u and v such that u∙v=a∙c and u+v=b
   This means u and v are factors of a∙c that when added together equal b
2. Rewrite the expression as (ax^2+ux)+(vx+c)
3. Factor out what you can from each parentheses
4. Factor out a common term
5. Check by multiplying the factors together (FOIL)

Here’s an example, where 1 is the leading coefficient (click here):

Factor\:x^2-5x+6

1. Find u and v such that u∙v=a∙c and u+v=b

a∙c=1∙6=6

6 can be written as the product of 1 and 6, -1 and -6, 3 and 2, or of -3 and -2. We need to pick the factors of 6 that equal -5. 

-3+(-2)=-5
-3∙-2=6
u=-3,\:v=-2

2. Rewrite the expression as (ax^2+ux)+(vx+c)

(x^2+(-3x))+(-2x+6)
(x^2-3x)+(-2x+6)

3. Factor out what you can from each parentheses

(x^2-3x)+(-2x+6)
x(x-3)+2(-x+3)
x(x-3)-2(x-3)

We factored out a -1 from (-x+3) on the last step because we want the expressions inside the parentheses to be the same for step 4.

4. Factor out a common term

x(x-3)-2(x-3)
(x-3)(x-2)

5 .Check by multiplying the factors together

(x-3)(x-2)=x^2-2x-3x+6=x^2-5x+6

Here’s an example when 1 isn’t the leading coefficient (click here):

Factor\:2x^2+x-6

1. Find u and v such that u∙v=a∙c and u+v=b

a∙c=2∙-6=-12

Factors of -12:    -12 and 1, 12 and -1, -6 and 2, 6 and -2, -4 and 3, 4 and -3
4-3=1
4∙-3=-12
u=-4,\:v=3

2. Rewrite the expression as (ax^2+ux)+(vx+c)

(2x^2-4x)+(3x-6)

3. Factor out what you can from each parentheses

(2x^2-4x)+(3x-6)
2x(x-2)+3(x-2)

4. Factor out a common term

(x-2)(2x+3)

5. Check by multiplying the factors together

(x-2)(2x+3)=2x^2+3x-4x-6=(2x)^2-x-6

The best advice for factoring quadratics is to practice factoring as many quadratics as you can. The more you practice factoring, the faster you will get. Soon, you’ll be able to skip the steps and factor it all in your head within seconds. For more help or practice on the topic, check out  Symbolab’s Practice.

Until next time,

Leah