How to factor $4n^2 - 35n + 49$ quickly

Key takeaway: This trinomial looks a lot like a perfect-square pattern, so it is worth checking that first. A fast recognition step can save time.

Step 1: Check for a perfect square

We have

$4n^2-35n+49$

Since $4n^2=(2n)^2$ and $49=7^2$, test whether the middle term matches the pattern

$(a-b)^2=a^2-2ab+b^2$

Here, $a=2n$ and $b=7$.

The middle term would be

$-2(2n)(7)=-28n$

That does not match $-35n$, so this is not a perfect square trinomial.

Step 2: Use factoring by grouping style

We need two numbers that multiply to $4\cdot 49=196$ and add to $-35$.

Those numbers are $-28$ and $-7$.

Step 3: Split the middle term

$4n^2-28n-7n+49$

Step 4: Group

$4n(n-7)-7(n-7)$

Step 5: Factor out the common binomial

$\boxed{(4n-7)(n-7)}$

Check

$(4n-7)(n-7)=4n^2-28n-7n+49=4n^2-35n+49$

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Pitfalls the pros know 👇 Do not stop after spotting $4n^2$ and $49$; the expression is only a perfect square if the middle term is exactly $-2ab$. Here it is not, so forcing $(2n-7)^2$ would be wrong.

What if the problem changes? If the middle term had been $-28n$ instead of $-35n$, the factorization would collapse to $(2n-7)^2$. Small coefficient changes can completely change the factoring pattern, so always verify the middle term.

Tags: perfect square trinomial, factoring by grouping, quadratic factorization