1Cademy - Multiplying $$(5m - 9n)(5m + 9n)$$ Using the Product of Conjugates Pattern

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Product of Conjugates Pattern

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Multiplying $(5m - 9n)(5m + 9n)$ Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply $(5m - 9n)(5m + 9n)$ . Both binomials share the same first term $5m$ and the same last term $9n$ , with one using subtraction and the other addition, confirming they are conjugates. Use the formula $(a - b)(a + b) = a^2 - b^2$ , where $a = 5m$ and $b = 9n$ .

Step 1 — Square the first term, $5m$ : $(5m)^2 = 25m^2$ . Square the coefficient and the variable separately: $5^2 = 25$ and $m^2 = m^2$ .

Step 2 — Square the last term, $9n$ : $(9n)^2 = 81n^2$ . Square both parts: $9^2 = 81$ and $n^2 = n^2$ .

Step 3 — Write the difference of squares:

$(5m - 9n)(5m + 9n) = 25m^2 - 81n^2$

The product is $25m^2 - 81n^2$ . This example extends the pattern to conjugate pairs in which both terms involve different variables with numerical coefficients. Each term must be squared by applying the squaring operation to both its coefficient and its variable independently — for instance, $(5m)^2 = 5^2 \cdot m^2 = 25m^2$ , not $5m^2$ . The resulting difference of squares contains two different variables, which is a new feature compared to the single-variable examples.

Updated 2026-04-29

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