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Difference of Squares – Explanation & Examples

 

A quadratic equation is a polynomial of second degree usually in the form of f(x) = ax2 + bx + c where a, b, c, ∈ R and a ≠ 0. The term ‘a’ is referred to as the leading coefficient, while ‘c’ is referred to as the absolute term of f (x). Every quadratic equation has two values of the unknown variable usually known as the roots of the equation (α, β).

What is Difference of Squares?

The difference of two squares is a theorem that tells us if a quadratic equation can be written as a product of two binomials, in which one shows the difference of the square roots and the other shows the sum of the square roots. One thing to note about this theorem is that, it is not applicable to the SUM of squares.

Difference of Squares Formula

The difference of square formula is an algebraic form of equation that is used to express the differences between two square values. A difference of square is expressed in the form:

a2 – b2; where both the first and last term are perfect squares. Factoring the difference of the two squares, gives;

a2 – b2 = (a + b) (a – b)

This is true because, (a + b) (a – b) = a2 – ab + ab – b2 = a2 – b2

How to Factor Difference of Squares?

In this section, we are going to learn how to factorize algebraic expressions using the difference of square formula. To factor a difference of squares, the following steps are undertaken:

  • Check if the terms have the greatest common factor (GCF) and factor it out. Remember to include the GCF in your final answer
  • Determine the numbers that will produce the same results and apply the formula: a2– b2 = (a + b) (a – b) or (a – b) (a + b)
  • Check whether the remaining terms can be factored any further.

Let’s solve a few examples by applying these steps.

 

Example 1

Factor 64 – x2

Solution

Since we know the square of 8 is 64, then we can rewrite the expression as;
64 – x2 = (8)2 – x2
Now, apply the formula a2 – b2 = (a + b) (a – b) to factorize expression;
= (8 + x) (8 – x).

 

Example 2

Factorize
x 2 −16

Solution

Since x2−16 = (x) 2− (4)2, therefore apply the difference square formula a2 – b2 = (a + b) (a – b), where a and b in this case are x and 4 respectively.

Therefore, x2 – 42 = (x + 4) (x – 4)

Example 3

Factor 3a2 – 27b2

Solution

Since 3 is GCF of the terms, we factor it out.
3a2 – 27b2 = 3(a2 – 9b2)
=3[(a)2 – (3b)2]
Now apply a2 – b2 = (a + b) (a – b) to get;
= 3(a + 3b) (a – 3b)

 

Example 4

Factor x3 – 25x
Solution

Since the GCF = x, factor it out;
x3 – 25x = x (x2 – 25)
= x (x2 – 52)
Apply the formula a2 – b2 = (a + b) (a – b) to get;
= x (x + 5) (x – 5).

Example 5

Factor the expression (x – 2)2 – (x – 3)2

Solution

In this problem a = (x – 2) and b = (x – 3)

We now apply a2 – b2 = (a + b) (a – b)

= [(x – 2) + (x – 3)] [(x – 2) – (x – 3)]

= [x – 2 + x – 3] [x – 2 – x + 3]

Combine the like terms and simplify the expressions;

[x – 2 + x – 3] [x – 2 – x + 3] = > [2x – 5] [1]

= [2x – 5]

 

Example 6

Factor the expression 25(x + y)2 – 36(x – 2y)2.

Solution

Rewrite the expression in the form a2 – b2.

25(x + y)2 – 36(x – 2y)2 => {5(x + y)}2 – {6(x – 2y)}2
Apply the formula a2 – b2 = (a + b) (a – b) to get,

= [5(x + y) + 6(x – 2y)] [5(x + y) – 6(x – 2y)]

= [5x + 5y + 6x – 12y] [5x + 5y – 6x + 12y]

Collect like terms and simplify;

= (11x – 7y) (17y – x).

 

Example 7

Factor 2x2– 32.

Solution

Factor out the GCF;
2x2– 32 => 2(x2– 16)
= 2(x2 – 42)

Applying the difference squares formula, we get;
= 2(x + 4) (x – 4)

 

Example 8

Factor 9x6 – y8

Solution

First, rewrite 9x6 – y8 in the form a2 – b2.

9x6 – y8 => (3x3)2 – (y4)2

Apply a2 – b2 = (a + b) (a – b) to get;

= (3x3 – y4) (3x3 + y4)

 

Example 9

Factor the expression 81a2 – (b – c)2

Solution

Rewrite 81a2 – (b – c)2 as a2 – b2
= (9a)2 – (b – c)2
By applying the formula of a2 – b2 = (a + b) (a – b) we get,
= [9a + (b – c)] [9a – (b – c)]
= [9a + b – c] [9a – b + c ]

 

Example 10

Factor 4x2– 25

Solution

= (2x)2– (5)2
= (2x + 5) (2x – 5

 

Practice Questions

Factorize the following algebraic expressions:

  1. y2– 1
  2. x2– 81
  3. 16x 4 – 1
  4. 9x 3 – 81x
  5. 18x 2 – 98y2
  6. 4x2 – 81
  7. 25m2 -9n2
  8. 1 – 4z2
  9. x4– y4
  10. y4 -144

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