1Cademy - Factoring $$64m^2 + 112mn + 49n^2$$

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Perfect Square Trinomials Pattern

Example

Factoring $64m^2 + 112mn + 49n^2$

Factor the two-variable trinomial $64m^2 + 112mn + 49n^2$ by applying the perfect square trinomials pattern.

Step 1: Check if the trinomial fits the perfect square pattern $a^2 + 2ab + b^2$ .

Is the first term a perfect square? Yes: $64m^2 = (8m)^2$ , so $a = 8m$ .
Is the last term a perfect square? Yes: $49n^2 = (7n)^2$ , so $b = 7n$ .
Is the middle term $2ab$ ? Check: $2(8m)(7n) = 112mn$ . Yes, the middle term matches.

Step 2: Write the expression as the square of a binomial. Because the middle term is positive, use the pattern $(a + b)^2$ : $64m^2 + 112mn + 49n^2 = (8m + 7n)^2$

Step 3: Check by multiplying: $(8m + 7n)^2 = (8m)^2 + 2(8m)(7n) + (7n)^2 = 64m^2 + 112mn + 49n^2$ ✓

The factored form is $(8m + 7n)^2$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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