1Cademy - Squaring $$(2x - 3y)^2$$ Using the Binomial Squares Pattern

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Binomial Squares Pattern

Example

Squaring $(2x - 3y)^2$ Using the Binomial Squares Pattern

Apply the Binomial Squares Pattern to multiply $(2x - 3y)^2$ . Because the binomial involves subtraction, use the form $(a - b)^2 = a^2 - 2ab + b^2$ , where $a = 2x$ and $b = 3y$ .

Step 1 — Square the first term: $(2x)^2 = 4x^2$ . Both the coefficient and the variable are squared: $2^2 = 4$ and $x^2 = x^2$ .

Step 2 — Square the last term: $(3y)^2 = 9y^2$ . Again, square both parts: $3^2 = 9$ and $y^2 = y^2$ .

Step 3 — Double their product: $2 \cdot (2x) \cdot (3y) = 12xy$ . Because the binomial uses subtraction, the middle term is negative: $-12xy$ .

Assemble the three pieces into a trinomial:

$(2x - 3y)^2 = 4x^2 - 12xy + 9y^2$

This example extends the pattern to a binomial whose two terms contain different variables ( $x$ and $y$ ), each with a numerical coefficient greater than $1$ . When squaring the first and last terms, both the coefficient and the variable must be squared separately — for instance, $(2x)^2 = 2^2 \cdot x^2 = 4x^2$ , not $2x^2$ . The middle term $12xy$ involves the product of both variables, producing a two-variable trinomial.

Updated 2026-04-21

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OpenStax Elementary Algebra

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