Factoring Quadratic

In this lesson, we will learn how to factor quadratic equations that are the Perfect Square Trinomials (Square of a Sum or Square of a Difference) and Difference of Two Squares.

Related Topics:
Factoring Out Common Factors (GCF).
Factoring Quadratic Equations where the coefficient of x2 is 1.
Factoring Quadratic Equations where the coefficient of x2 is greater than 1 
Factoring Quadratic Equations by Completing the Square
Factoring Quadratic Equations using the Quadratic Formula.

 

 

In some cases, recognizing some common patterns in the quadratic equation will help you to factorize the quadratic. 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.

Perfect Square Trinomial (Square Of a Sum)

A square of a sum or perfect square trinomial is a type of quadratic equations of the form:

x2 + 2bx + b2 = (x + b)2

Example 1: x2 + 2x + 1 = 0
  (x + 1)2 = 0
  x= -1
   
Example 2:

x2 + 6x + 9 = 0

  x2 + 2(3)x + 32 = 0
  (x + 3)2 = 0
  x = -1

Perfect Square Trinomial (Square of a Difference)

A square of difference is a type of quadratic equations of the form:

x2 – 2bx + b2 = (xb)2

Example 1: x2 – 2x + 1 = 0
  (x – 1)2 = 0
  x=1
   
Example 2:

x2 – 6x + 9 = 0

  x2 – 2(3)x + 32 = 0
  (x – 3)2 = 0
  x 3 = 0 ⇒ x = 3

 

 

This video provides examples of how to factor and solve quadratic equations when the equations are perfect square trinomials.

 

 

Perfect Square Trinomials
One special case when trying to factor polynomials is a perfect square trinomial. Unlike a difference of perfect squares, perfect square trinomials are the result of squaring a binomial. It's important to recognize the form of perfect square trinomials so that we can easily factor them without going through the steps of factoring trinomials, which can be very time consuming.



Factor perfect square trinomial

 

 

Difference of Two Squares

A difference of two squares is a type of quadratic equations of the form:

(a + b)(a – b) = a2b2

Example: x2 – 25 = 0
  x2 – 52 = 0
  (x + 5)(x – 5) = 0
   

We get two values for x:

  x = -5, x = 5

Be careful! This method only works for difference of two squares and not for the sum of two squares: a2 + b2 ≠ (a + b)(ab)

 

 

The following videos explain how to factor a difference of squares.

 

 

 

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