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Difference of squares

The difference of squares used backwards factorises a difference of two squares. You do not need the quadratic formula: the square roots of the two terms become the sum and the difference. A sum of two squares, such as x2+25x^2 + 25, cannot be factorised this way over the real numbers. Always check by expanding again or by substituting a number.

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)difference of squares backwards
x2−25=(x+5)(x−5)x^2 - 25 = (x + 5)(x - 5)example with b=5b = 5

Symbols

a, ba,\ bnumbers or expressions

Example

x2−25=(x+5)(x−5)x^2 - 25 = (x + 5)(x - 5). For x=7x = 7: 49−25=2449 - 25 = 24 and 12⋅2=2412\cdot 2 = 24.

Look for two squares with a minus sign between them. A plus sign does not give the same factorisation.
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Part of Mathematics 1T: Factorisation and polynomial division.