1Cademy - Solving $$(x - 3)^2 = 16$$ Using the Square Root Property

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Square Root Property

Example

Solving $(x - 3)^2 = 16$ Using the Square Root Property

Solve $(x - 3)^2 = 16$ by applying the Square Root Property to a squared binomial. The entire expression $(x - 3)$ is treated as the quadratic term.

Apply the Square Root Property. Since the squared binomial is already isolated:

$x - 3 = \pm\sqrt{16}$

Simplify. Because $16$ is a perfect square ( $4^2 = 16$ ):

$x - 3 = \pm 4$

Write as two separate equations and solve each:

$x - 3 = 4 \implies x = 7$

$x - 3 = -4 \implies x = -1$

Check both solutions by substituting into the original equation:

For $x = 7$ : $(7 - 3)^2 = 4^2 = 16$ ✓

For $x = -1$ : $(-1 - 3)^2 = (-4)^2 = 16$ ✓

The solutions are $x = 7$ and $x = -1$ . This example demonstrates that the Square Root Property applies to any expression of the form $(\text{binomial})^2 = k$ , not just $x^2 = k$ . When the squared expression is a binomial, applying the property produces an equation of the form $x - h = \pm\sqrt{k}$ , and a final step of adding $h$ to both sides isolates the variable.

Updated 2026-04-21

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OpenStax Elementary Algebra

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