Example

Factoring 18w2−39w+1818w^2 - 39w + 18 Using the ac Method

To factor the trinomial 18w2−39w+1818w^2 - 39w + 18 using the 'ac' method, begin by factoring out the greatest common factor (GCF). The terms 18w218w^2, −39w-39w, and 1818 share a GCF of 33. Factoring it out leaves 3(6w2−13w+6)3(6w^2 - 13w + 6). Next, use the 'ac' method on the trinomial inside the parentheses, where a=6a = 6 and c=6c = 6. Calculate the product ac=6⋅6=36ac = 6 \cdot 6 = 36. Find two numbers that multiply to 3636 and add up to the middle coefficient, −13-13. These numbers are −4-4 and −9-9 (since −4⋅(−9)=36-4 \cdot (-9) = 36 and −4+(−9)=−13-4 + (-9) = -13). Rewrite the middle term −13w-13w as −4w−9w-4w - 9w, giving 3(6w2−4w−9w+6)3(6w^2 - 4w - 9w + 6). Factor by grouping: 3[2w(3w−2)−3(3w−2)]3[2w(3w - 2) - 3(3w - 2)] to obtain 3(2w−3)(3w−2)3(2w - 3)(3w - 2). Verify by multiplying the factors: 3(2w−3)(3w−2)=3(6w2−4w−9w+6)=3(6w2−13w+6)=18w2−39w+183(2w - 3)(3w - 2) = 3(6w^2 - 4w - 9w + 6) = 3(6w^2 - 13w + 6) = 18w^2 - 39w + 18. The factored form is 3(2w−3)(3w−2)3(2w - 3)(3w - 2).

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Updated 2026-06-05

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