# Factoring Quadratic Equations - Common Factors

In this algebra lesson, we will discuss how factoring can be used to solve Quadratic Equations, which are equations of the form:
ax2 + bx + c = 0 where a, b and c are numbers and a ≠ 0.

The simplest way to factoring quadratic equations would be to find common factors. Sometimes, the first step is to factor out the greatest common factor before applying other factoring techniques.

In some cases, recognizing some common patterns in the equation will help you to factorize the quadratic equation. For example, the quadratic equation could be a Perfect Square Trinomial (Square of a Sum or Square of a Difference) or Difference of Two Squares.

In other cases, you will have to try out different possibilities to get the right factors for quadratic equations. This is still manageable if the coefficient of x2 is 1. If the coefficient of x2 is greater than 1 then you may want to consider using the Quadratic formula.

### Factoring Out Common Factors

We can factorize quadratic equations by looking for values that are common.

 Example: x2 + 3x = 0 We find that the two terms have x in common. We “take out” x from each term. x(x + 3) = 0 We have two factors when multiplied together gets 0. We know that any number multiplied by 0 gets 0. So, either one or both of the terms are 0 i.e. x = 0 or x + 3 = 0 ⇒ x = -3       isolate variable x

This tells us that the quadratic equation x2 + 3x = 0 can have two values (two solutions) for x which are x = 0 or x = –3

How to solve a quadratic equation by factoring out the greatest common factor?

Examples:
Solve
x2 + 4x = 0
14x2 - 35x = 0

Factoring and solving quadratic equations by factoring out the greatest common factor

Examples:
Solve
12x2 + 18x = 0
4x2 = 20x

How to factor and solve quadratic equations when the first step is to factor out the greatest common factor before apply other factoring techniques?

Examples:
Factor
5x2 - 45 = 0
3x2 - 6x - 45 = 0

Factoring Completely
The first step in all factoring problems is to check to see if a Greatest Common Factor can be factored out of the original polynomial.

Example:
Factor
3x2 + 6x - 24

Factoring Out The Greatest Common Factor
To find the GCF of a polynomial

1. Write each term in factored form.
2. Identify the factors common to all terms.
3. Factor out the GCF.

Examples:
Factor out the GCF

1. 2x4 - 16x2
2. 4x2y3 + 20xy2 + 12xy
3. -2x3 + 8x2 - 4x
4. -y3 - 2y2 + y - 7

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.