Solved Examples: Difference of Squares

Solved Example 1

Factor `x^2 - 4`

Solution

Rewrite 4 as `2^2`

`x^2 – 4 = x^2 – 2^2`

Comparing with `a^2 – b^2`, we get `a = x` and `b = 2`. Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`x^2 – 2^2 = (x + 2)(x – 2)`

Solved Example 2

Factor `x^4 – 81`

Solution

Rewrite `x^4` as `(x^2)^2` and 81 as `9^2`. Thus, we get

`x^4 – 81 = (x^2)^2 – 9^2`

Comparing with `a^2 – b^2`, we get `a = x^2` and `b = 9`. Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`(x^2)^2 – 9^2 = (x^2 + 9)(x^2 – 9)`

Rewrite 9 as `3^2`

`(x^2 + 9)(x^2 – 9) = (x^2 + 9)(x^2 – 3^2)`

Again applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`(x^2 + 9)(x^2 – 3^2) = (x^2 + 9)(x + 3)(x – 3)`

Solved Example 3

Factor `2x^2 – 72`

Solution

Factor out 2 from the given expression,

`= 2(x^2 – 36)`

Rewrite 36 as `6^2`.

` = 2(x^2 – 6^2)`

Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`= 2(x + 6)(x – 6)`

Solved Example 4

Factor `a^2 – b^4`

Solution

Rewrite `b^4` as `(b^2)^2`  to get

`a^2 – (b^2)^2`

Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`(a + b^2)(a – b^2)`

Solved Example 5

Factor `25x^2 – 36y^2`

Solution

Rewrite 25 as `5^2` and 36 as `6^2`.

`(5x)^2 – (6y)^2`

Applying the formula `a^2 – b^2 = (a + b)(a – b)`, we get

`(5x + 6y)(5x – 6y)`

Solved Example 6

Factor `32x^3 – 50x`

Solution

Factor out `2x` from the given expression,

`= 2x(16x^2 – 25)`

Rewrite 16 as `4^2` and 25 as `5^2`

`= 2x((4x)^2 – 5^2)`

Applying the formula `a^2 – b^2 = (a + b)(a – b)`,

`= 2x(4x + 5)(4x – 5)`