1Cademy - Factoring $$36x^2 + 84xy + 49y^2$$

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Perfect Square Trinomials Pattern

Example

Factoring $36x^2 + 84xy + 49y^2$

Factor $36x^2 + 84xy + 49y^2$ by recognizing it as a perfect square trinomial in two variables.

Step 1 — Check whether the trinomial fits the pattern $a^2 + 2ab + b^2$ :

Is the first term a perfect square? Yes: $36x^2 = (6x)^2$ , so $a = 6x$ .
Is the last term a perfect square? Yes: $49y^2 = (7y)^2$ , so $b = 7y$ .
Is the middle term $2ab$ ? Check: $2 \cdot 6x \cdot 7y = 84xy$ . Yes, the middle term is $84xy$ . ✓

Step 2 — Write the square of the binomial: Since the trinomial matches the addition pattern with $a = 6x$ and $b = 7y$ :

$36x^2 + 84xy + 49y^2 = (6x)^2 + 2(6x)(7y) + (7y)^2 = (6x + 7y)^2$

Step 3 — Check by multiplying: $(6x + 7y)^2 = (6x)^2 + 2 \cdot 6x \cdot 7y + (7y)^2 = 36x^2 + 84xy + 49y^2$ ✓.

The factored form is $(6x + 7y)^2$ . This example extends the Perfect Square Trinomials Pattern to a trinomial with two variables. Both the first and last terms involve squaring a monomial with a variable — $(6x)^2 = 36x^2$ and $(7y)^2 = 49y^2$ — and the middle term $84xy$ contains the product of both variables. Recognizing the perfect square structure allows direct factoring into the square of a two-variable binomial.

Updated 2026-04-21

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