Example

Factoring 16x2−32x+1216x^2 - 32x + 12 Using the ac Method

To factor the trinomial 16x2−32x+1216x^2 - 32x + 12 using the 'ac' method, first identify and extract the greatest common factor (GCF). The terms 16x216x^2, −32x-32x, and 1212 share a GCF of 44. Factoring it out yields 4(4x2−8x+3)4(4x^2 - 8x + 3). Now, apply the 'ac' method to the remaining trinomial where a=4a = 4 and c=3c = 3. The product ac=4⋅3=12ac = 4 \cdot 3 = 12. We need two numbers that multiply to 1212 and add to the middle coefficient, −8-8. These numbers are −2-2 and −6-6 (since −2⋅(−6)=12-2 \cdot (-6) = 12 and −2+(−6)=−8-2 + (-6) = -8). Split the middle term −8x-8x using these numbers: 4(4x2−2x−6x+3)4(4x^2 - 2x - 6x + 3). Next, factor by grouping: 4[2x(2x−1)−3(2x−1)]4[2x(2x - 1) - 3(2x - 1)] which simplifies to 4(2x−3)(2x−1)4(2x - 3)(2x - 1). Check the result by multiplying: 4(2x−3)(2x−1)=4(4x2−2x−6x+3)=4(4x2−8x+3)=16x2−32x+124(2x - 3)(2x - 1) = 4(4x^2 - 2x - 6x + 3) = 4(4x^2 - 8x + 3) = 16x^2 - 32x + 12. The completely factored expression is 4(2x−3)(2x−1)4(2x - 3)(2x - 1).

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Updated 2026-06-17

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