1Cademy - Solving $$(x - \frac{1}{2})^2 = \frac{5}{4}$$ Using the Square Root Property

Learn Before

Square Root Property

Example

Solving $(x - \frac{1}{2})^2 = \frac{5}{4}$ Using the Square Root Property

Solve $(x - \frac{1}{2})^2 = \frac{5}{4}$ by applying the Square Root Property to a squared binomial that involves fractions. This example combines the Square Root Property with the Quotient Property of Square Roots to simplify the radical of a fraction.

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

$x - \frac{1}{2} = \pm\sqrt{\frac{5}{4}}$

Rewrite the radical as a fraction of square roots using the Quotient Property:

$x - \frac{1}{2} = \pm\frac{\sqrt{5}}{\sqrt{4}}$

Simplify the radical in the denominator. Since $4$ is a perfect square ( $2^2 = 4$ ), $\sqrt{4} = 2$ :

$x - \frac{1}{2} = \pm\frac{\sqrt{5}}{2}$

Solve for $x$ by adding $\frac{1}{2}$ to both sides:

$x = \frac{1}{2} \pm \frac{\sqrt{5}}{2}$

Rewrite as two solutions:

$x = \frac{1}{2} + \frac{\sqrt{5}}{2}, \quad x = \frac{1}{2} - \frac{\sqrt{5}}{2}$

When the constant on the right side of a squared-binomial equation is a fraction, applying the Square Root Property produces the square root of a fraction. The Quotient Property separates this into a fraction of two individual square roots, and if the denominator is a perfect square, it simplifies to an integer. Because $5$ has no perfect square factors, $\sqrt{5}$ remains in radical form, and the final solutions are expressed as sums and differences of fractions involving radicals.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After