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Factoring formulas

Difference of two squares

x² - a² = (x + a)(x - a)
x² - a²=(x + a)(x - a)recognise the pattern, then replace it with the factor form

When two squared terms are subtracted, they factor into matching brackets with opposite signs.

Detailed notes

Formula Walkthrough

These identities work because each factor form expands back into the expression on the left.

Step 1: Difference of two squares

When the middle terms cancel, only the two squares remain.

(x + a)(x - a) = x2 - ax + ax - a2
x2 - a2 = (x + a)(x - a)

Step 2: Perfect square with plus

The bracket is repeated, so the middle term appears twice.

(x + a)2 = (x + a)(x + a)
x2 + 2ax + a2 = (x + a)2

Step 3: Perfect square with minus

The same repeated-bracket idea works with x - a, but the middle term is negative.

(x - a)2 = (x - a)(x - a)
x2 - 2ax + a2 = (x - a)2

Step 4: General two-number trinomial

The x coefficient comes from adding the two bracket numbers. The constant comes from multiplying them.

(x + a)(x + b) = x2 + ax + bx + ab
x2 + (a + b)x + ab = (x + a)(x + b)