1Cademy - Simplifying $$(1 + 3\sqrt{x})^2$$

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Binomial Squares Pattern

Example

Simplifying $(1 + 3\sqrt{x})^2$

Apply the Binomial Squares Pattern to square a binomial that includes a variable inside a square root.

Use the addition form $(a + b)^2 = a^2 + 2ab + b^2$ , where $a = 1$ and $b = 3\sqrt{x}$ .

Step 1 — Square the first term: $1^2 = 1$ .

Step 2 — Double the product of the two terms: $2 \cdot 1 \cdot 3\sqrt{x} = 6\sqrt{x}$ .

Step 3 — Square the last term: $(3\sqrt{x})^2 = 9x$ . Square the coefficient and the radical separately: $3^2 = 9$ and $(\sqrt{x})^2 = x$ .

Assemble the three pieces:

$(1 + 3\sqrt{x})^2 = 1 + 6\sqrt{x} + 9x$

No like terms remain to combine, so the result stays as a three-term expression. This example extends the Binomial Squares Pattern to a binomial whose second term is a coefficient times a variable square root. The critical observation is that $(3\sqrt{x})^2 = 9x$ , not $3x$ — both the numerical coefficient and the radical must be squared independently.

Updated 2026-04-21

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OpenStax Elementary Algebra

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