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How do I Factor Quadratic Equations?

A Quadratic Function in Standard Form :

x^2+bx+c

In Factored Form it looks like this:

(x+r)*(x+s)

where r,s are the 2 Zeros.

When distributing we get:

x^2+2 r s + r s

Matching the Coefficients

x^2+bx+c = x^2 + (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

Ax^2+Bx+C=0

with

A\ne 1

.
We first divide the entire equation by A to get:

x^2+(B/A)x+C/A = 0

Setting b=B/A and c=C/A we rewrite as

x^2+bx+c=0

The Factored Form looks like this:

(x+r)*(x+s) = 0

– r,s are the 2 Zeros.

Distributing terms we get

(x^2+2 r s + r s) = 0

We again Match the Coefficients:

x^2+bx+c = x^2 + (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
$x^2- 6x+8$ .
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:
$x^2- 6x+8 = (x-4)*(x-2)$ .

When asked to solve the Quadratic Equation
$x^2- 6x+8=0$ .

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
$x^2-6x+8$ we must enter the 3 coefficients as
$a=1, b=-6$ and $c=8$ .

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

Since
$x^2- 6x+8 = (x-4)*(x-2)$

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

as the factored form.

Tip: When using the above Factor Quadratic Equation Solver to factor
$2x^2-12x+16$
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: