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

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Perfect Square Trinomials Pattern

Example

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

Factor the two-variable trinomial $49x^2 + 84xy + 36y^2$ 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: $49x^2 = (7x)^2$ , so $a = 7x$ .
Is the last term a perfect square? Yes: $36y^2 = (6y)^2$ , so $b = 6y$ .
Is the middle term $2ab$ ? Check: $2(7x)(6y) = 84xy$ . 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 $(a + b)^2$ : $49x^2 + 84xy + 36y^2 = (7x + 6y)^2$