Algebra

How to Factor a Quadratic Equation and Solve It.

Published by
Mike Woodhill PhD

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:

Mike Woodhill PhD

As a long time instructor I would like you to benefit from my Step by Step Calculators. You may also schedule a live tutoring session to succeed in your class. 😉

Published by
Mike Woodhill PhD

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