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.

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Related Pages
Factoring Out Common Factors (GCF)
Factoring Quadratic Equations (Square of a sum, Square of a difference, Difference of 2 squares
Factoring Quadratic Equations where the coefficient of x2 is greater than 1
Factoring Quadratic Equations by Completing the Square
Solving Quadratic Equations using the Quadratic Formula
More Lessons for Algebra
Math Worksheets

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?

x2 + 4x = 0
14x2 - 35x = 0

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

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?

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.

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.

Factor out the GCF

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

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