Identities and general quadratic factors

Identities describe every allowed input

Distribution gives (a + b)² = (a + b)(a + b) = a² + 2ab + b². Similarly (a-b)² = a²-2ab + b², and (a + b)(a-b) = a²-b². The middle terms of the last product cancel; the middle terms of a square combine instead.

For example, (x-3)² = x²-6x + 9, while (x-3)(x + 3) = x²-9. A square of a difference is not the difference of the squares.

Factor a quadratic whose leading coefficient is not one

For 6x² + 7x + 2, look for two terms whose product is 6 × 2 = 12 and sum is 7: 3 and 4. Split the middle term:

Check by expanding: 6x² + 3x + 4x + 2. Grouping works because the same complete factor occurs in both grouped terms. A common overall factor should be extracted first when present.

Check your understanding 1 Verify the cross terms

Factor 6x²-x-2 and 9y²-16.

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6x²-x-2 = (3x-2)(2x + 1), since the cross terms are 3x-4x = -x. Also 9y²-16 = (3y-4)(3y + 4) by the difference-of-squares identity.

A plausible factor pair can still fail

Does (2x + 1)(3x + 2) factor 6x² + 5x + 2?