What you will learn in this mini lesson

You will learn how to factor and solve Quadratic Equations in Standard Form when the leading coefficient A=1 and also when A ≠ 1. Our Step by Step Calculator allows you to factor and solve your own quadratic equation.

Quick Example when A ≠ 1
2x² -12x+16 Factor out 2:
= 2(x² -6x+8)
= 2(x-4)*(x-2) since (-4) + (-2) = -6 and (-4)*(-2)=8
Solving (x-4)=0 and (x-2)=0 yields the two zeros
x=4 and x=2 . That’s all 😉

How do I Factor Quadratic Equations?

A Quadratic Function in Standard Form :
x²+bx+c

In Factored Form it looks like this:
(x+r)*(x+s) where r,s are the 2 Zeros.

When distributing we get:
x²+(r + s)*x + r*s

Matching the Coefficients on both sides
x²+bx+c = x² + (r + s)*x + r*s

shows that the 2 Zeros r and s have to fulfill the 2 conditions:
1) r+s = b and
2) r*s = c

In Words:
1) r and s have to add to the value of the middle coefficient b.
2) r and s multiplied have to equal the constant coefficient c.

What if the leading coefficient A is not 1 ?

Let’s factor Ax²+Bx+C=0 with A ≠ 1 .
We first divide the entire equation by A to get:
x²+(B/A)x+C/A = 0

Setting b=B/A and c=C/A we rewrite as
x²+bx+c=0

The Factored Form looks like this:
(x+r)*(x+s) = 0 – r,s are the 2 Zeros.

Distributing terms we get
(x²+(r + s)*x + r*s) = 0

We again Match the Coefficients:
x²+bx+c = x² + (r + s)*x + r*s

It shows that the 2 Zeros r and s have to fulfill these 2 conditions:
1) r+s = b = B/A and
2) r s = c = C/A

In Words:
The 2 zeros r and s have to add to b = B/A.
And when multiplied equal c = C/A.
See below’s examples.

Sample Problem: How to Factor a Quadratic Equation?

1) Factor Quadratic Equations with Leading coefficient A = 1
We are to factor the Quadratic Equation
x²– 6x+8 .
The 2 zeros when multiplied have to equal 8.
That could be 8 and 1 OR 4 and 2, and their negatives.

Additionally, they have to add to -6 which implies
the 2 zeros must be -4 and -2.

Therefore, the factored version is:
x²– 6x+8 = (x-4)*(x-2) .

When asked to solve the Quadratic Equation
x²– 6x+8=0 .

we use the above factored version and set each factor equal to 0:
Since x-4=0 we get x=4 ,
and since x-2=0 we get x=2 .

Thus, the 2 zeros are x=4 , x=2

Easy, wasn’t it?

Tip: When using the above Factor Quadratic Equation Solver to factor
x²-6x+8 we must enter the 3 coefficients as
a=1, b=-6 and c=8.

2) Factor Quadratic Equations when A ≠ 1
We are to factor the Quadratic Equation
2x²– 12x+16 .
First divide by 2 to have a leading coefficient coefficient of A=1.
We get x²– 6x+8 as we had in the above example.

Since
x²– 6x+8 = (x-4)*(x-2)

we multiply by A=2 to get
2x²– 12x+16 = 2*(x-4)*(x-2)

as the factored form.

Tip: When using the above Factor Quadratic Equation Solver to factor
2x²-12x+16
we must enter the 3 coefficients a,b,c as
a=2, b=-12 and c=16.

This Video gives a great explanation on how to factor quadratic equations when the leading coefficient is not 1: