1Cademy - Factoring $$121x^2

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Difference of Squares Pattern

Example

Factoring $121x^2 - 49y^2$

Factor $121x^2 - 49y^2$ by recognizing it as a two-variable difference of squares and applying the pattern $a^2 - b^2 = (a - b)(a + b)$ .

Step 1 — Is this a difference of squares? Yes. Both terms are perfect squares involving different variables:

$121x^2 = (11x)^2$ , so $a = 11x$ .
$49y^2 = (7y)^2$ , so $b = 7y$ .

Rewrite the expression to show the squared structure: $(11x)^2 - (7y)^2$ .

Step 2 — Factor as the product of conjugates. Apply the difference of squares pattern with $a = 11x$ and $b = 7y$ :

$121x^2 - 49y^2 = (11x - 7y)(11x + 7y)$

Step 3 — Check by multiplying: $(11x - 7y)(11x + 7y) = 121x^2 + 77xy - 77xy - 49y^2 = 121x^2 - 49y^2$ ✓

The factored form is $(11x - 7y)(11x + 7y)$ . This example extends the Difference of Squares Pattern to a binomial in which both terms contain a different variable with a numerical coefficient — $11x$ and $7y$ . Recognizing that $121 = 11^2$ and $49 = 7^2$ is the key step; once both perfect-square structures are identified, the conjugate factors follow directly.

Updated 2026-04-21

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OpenStax Elementary Algebra

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