How to factor 3n^2-8n+4

Key takeaway: When the leading coefficient is not 1, the AC method is often the cleanest path. You are looking for a pair that fits both the product and the sum, then grouping does the rest.

We factor

$3n^2-8n+4$

Step 1: Multiply $a\cdot c$

$3\cdot 4=12$

Step 2: Find two numbers

We need two numbers that multiply to $12$ and add to $-8$.

Those numbers are $-6$ and $-2$.

Step 3: Split the middle term

$3n^2-6n-2n+4$

Step 4: Group

$3n(n-2)-2(n-2)$

Step 5: Factor the common binomial

$\boxed{(3n-2)(n-2)}$

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Pitfalls the pros know 👇 A common error is pairing $-4$ and $-3$ because they multiply to $12$, but they do not add to $-8$. Always check both conditions before moving on.

What if the problem changes? If the middle term were $+8n$, the factorization would change to $(3n-2)(n+2)$ after rechecking the sign pattern. One sign change can flip the entire setup.

Tags: AC method, factor by grouping, trinomial