Difference of two squares

Whenever we have an expression in the form of:

a^2 - b^2

We can factorise it into:

(a - b)(a + b)

Similarly, if we have an expression in the form of:

(a - b)(a + b)

We can expand it into:

a^2 - b^2

This has a lot of uses. For example, it’s a key step in the method of rationalising the denominator of a fraction.

Examples

Example: Factorise x^2 - 9

• We know that a^2 - b^2 factorises to (a - b)(a + b).
• Here, a = x and b = 3 (since 3^2 = 9).
• So, x^2 - 9 factorises to (x - 3)(x + 3).

Answer: (x - 3)(x + 3)

Example: Expand (x - 5)(x + 5)

• We know that (a - b)(a + b) expands to a^2 - b^2.
• Here, a = x and b = 5.
• So, (x - 5)(x + 5) expands to x^2 - 25.

Answer: x^2 - 25

Example: Factorise 4x^2 - 1

• We know that a^2 - b^2 factorises to (a - b)(a + b).
• Here, a = 2x (since (2x)^2 = 4x^2) and b = 1 (since 1^2 = 1).
• So, 4x^2 - 1 factorises to (2x - 1)(2x + 1).

Answer: (2x - 1)(2x + 1)

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