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 A Quadratic Function in Standard Form : In Factored Form it looks like this: When distributing we get: Matching the Coefficients on both sides shows that the 2 Zeros r and s have to fulfill the 2 conditions: In Words: Let’s factor Ax²+Bx+C=0 with A ≠ 1 . Setting b=B/A and c=C/A we
rewrite as What you will learn in this mini lesson
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?
x2+bx+c
(x+r)*(x+s) where r,s are the 2 Zeros.
x2+(r + s)*x + r*s
x2+bx+c = x2 + (r + s)*x +
r*s
1) r+s = b and
2) r*s = c
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 ?
We first divide the entire equation by A to get:
x²+(B/A)x+C/A = 0
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:
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What is factoring?
A polynomial with rational coefficients can sometimes be written as a product of lower-degree polynomials that also have rational coefficients. In such cases, the polynomial is said to "factor over the rationals." Factoring is a useful way to find rational roots (which correspond to linear factors) and simple roots involving square roots of integers (which correspond to quadratic factors).
Polynomials with rational coefficients always have as many roots, in the complex plane, as their degree; however, these roots are often not rational numbers. In such cases, the polynomial will not factor into linear polynomials.
Rational functions are quotients of polynomials. Like polynomials, rational functions play a very important role in mathematics and the sciences. Just as with rational numbers, rational functions are usually expressed in "lowest terms." For a given numerator and denominator pair, this involves finding their greatest common divisor polynomial and removing it from both the numerator and denominator.