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: