Difference of two squares is a method of factoring certain kinds of problems. Anytime when subtracting one square from another, use this method. First recognize when it can be used.

 

When the original problem is something of the form

(x² – y²) then the answer will be in the form (x - y) (x + y)

 

Example

 

Factor x² – 9

 

This is an example of a difference of two squares. The first square is x² and the second is 9. The square root of each of them is what is used. The square root of x² is x and the square root of 9 is 3. Therefore, plug x and 3 into the formula.

(x - 3) (x + 3)

 

Answer!

 

Note- In the previous problem 9 is considered a square because it is a perfect square (refer to the section on perfect squares) because the square root of 9 is 3. If there had been a number like 8, it cannot be called a perfect square because its square root is not easily known.

 

 

 

 

 

Example

 

Factor x² – 49 The first square in this case is x² and the second is 49. The square root of each is x and 7. Put it in the correct form

(x – 7) (x + 7)

 

Answer!

 

 

 

 

 

 

Example

 

Factor a² – b² The first square root is a, and the second is b.

(a – b) (a + b)

 

Answer!

 

 

 

 

 

 

Example

 

Factor 4x² – 25

 

The first square root is 2x because the square root of 4 is 2. The second square root is 5.

(2 x – 5) (2 x + 5)

 

Answer!

 

 

 

 

 

 

 

Example

 

Factor (a + 2)² –64y ²

 

The first square root is a + 2 and the second is 8 y.

(a + 2 –8y) (a + 2 + 8y)

 

Rewrite in the correct order.

(a –8y + 2) (a + 8y + 2)

 

Answer!