1Cademy - Solving $$(x - 2)^2 + 3 = 30$$ Using the Square Root Property

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Square Root Property

Example

Solving $(x - 2)^2 + 3 = 30$ Using the Square Root Property

Solve $(x - 2)^2 + 3 = 30$ by first isolating the squared binomial and then applying the Square Root Property. Unlike earlier examples where the binomial square was already isolated, this equation has an additional constant term that must be removed before the property can be used.

Isolate the binomial term. Subtract $3$ from both sides:

$(x - 2)^2 = 27$

Apply the Square Root Property:

$x - 2 = \pm\sqrt{27}$

Simplify the radical. The largest perfect square factor of $27$ is $9$ . Rewrite $27$ as $9 \cdot 3$ and apply the Product Property:

$\sqrt{27} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}$

So $x - 2 = \pm 3\sqrt{3}$ .

Solve for $x$ by adding $2$ to both sides:

$x = 2 \pm 3\sqrt{3}$

Rewrite as two solutions:

$x = 2 + 3\sqrt{3}, \quad x = 2 - 3\sqrt{3}$

This example demonstrates that when a squared-binomial equation has additional terms on the same side as the binomial, those terms must be isolated first — much like Step 1 in the general procedure requires isolating the quadratic term before applying the Square Root Property. The resulting radical $\sqrt{27}$ is not in simplified form and must be reduced using the Product Property of Square Roots.

Updated 2026-04-21

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OpenStax Elementary Algebra

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