1Cademy - Choosing the Appropriate Special Product Pattern for $$(2x - 3)(2x + 3)$$, $$(8x - 5)^2$$, $$(6m + 7)^2$$, and $$(5x

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Product of Conjugates Pattern

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Choosing the Appropriate Special Product Pattern for $(2x - 3)(2x + 3)$ , $(8x - 5)^2$ , $(6m + 7)^2$ , and $(5x - 6)(6x + 5)$

Choose the correct pattern — Binomial Squares, Product of Conjugates, or general FOIL — for each product, then compute the result.

ⓐ $(2x - 3)(2x + 3)$ : The two binomials share the same first term $2x$ and the same last term $3$ , with one using subtraction and the other addition — they are conjugates. Apply the Product of Conjugates Pattern $(a - b)(a + b) = a^2 - b^2$ , with $a = 2x$ and $b = 3$ :

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

ⓑ $(8x - 5)^2$ : A single binomial is being squared, so use the Binomial Squares Pattern $(a - b)^2 = a^2 - 2ab + b^2$ , with $a = 8x$ and $b = 5$ :

$(8x - 5)^2 = (8x)^2 - 2(8x)(5) + 5^2 = 64x^2 - 80x + 25$

ⓒ $(6m + 7)^2$ : Again, a binomial is squared. Use $(a + b)^2 = a^2 + 2ab + b^2$ , with $a = 6m$ and $b = 7$ :

$(6m + 7)^2 = (6m)^2 + 2(6m)(7) + 7^2 = 36m^2 + 84m + 49$

ⓓ $(5x - 6)(6x + 5)$ : The two binomials do not share the same pair of first and last terms (the first terms $5x$ and $6x$ differ, and the last terms $6$ and $5$ differ), so neither special product pattern applies. Use FOIL:

$(5x - 6)(6x + 5) = 30x^2 + 25x - 36x - 30 = 30x^2 - 11x - 30$

The key skill is inspecting the expression before computing: a squared binomial signals the Binomial Squares Pattern, a product of conjugates signals the Conjugates Pattern, and anything else requires the general FOIL method.

Updated 2026-04-29

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OpenStax Elementary Algebra

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