1Cademy - Multiplying $$(2x + 5)(2x - 5)$$ Using the Product of Conjugates Pattern

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Product of Conjugates Pattern

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Multiplying $(2x + 5)(2x - 5)$ Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply $(2x + 5)(2x - 5)$ . Confirm the binomials are conjugates: both share the same first term $2x$ and the same last term $5$ , with one using addition and the other subtraction. Use the formula $(a + b)(a - b) = a^2 - b^2$ , where $a = 2x$ and $b = 5$ .

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

Step 2 — Square the last term, $5$ : $5^2 = 25$ .

Step 3 — Write the difference of squares:

$(2x + 5)(2x - 5) = 4x^2 - 25$

The product is $4x^2 - 25$ . This example extends the pattern to a conjugate pair whose variable term has a numerical coefficient greater than $1$ . When squaring a monomial like $2x$ , both the coefficient and the variable must be squared: $(2x)^2 = 2^2 \cdot x^2 = 4x^2$ , not $2x^2$ . A common mistake is to forget to square the coefficient, writing $2x^2$ instead of the correct $4x^2$ .

Updated 2026-04-29

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