1Cademy - Solving $$\frac{2}{3}u^2 + 5 = 17$$ Using the Square Root Property

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How to Solve a Quadratic Equation Using the Square Root Property

Example

Solving $\frac{2}{3}u^2 + 5 = 17$ Using the Square Root Property

Solve $\frac{2}{3}u^2 + 5 = 17$ by applying the four-step procedure for the Square Root Property. This equation requires subtracting a constant and multiplying by a reciprocal before the property can be used, and the resulting radical must be simplified.

Step 1 — Isolate the quadratic term and make its coefficient one. Subtract $5$ from both sides:

$\frac{2}{3}u^2 = 12$

The coefficient of $u^2$ is $\frac{2}{3}$ . Multiply both sides by the reciprocal $\frac{3}{2}$ :

$\frac{3}{2} \cdot \frac{2}{3}u^2 = \frac{3}{2} \cdot 12$

$u^2 = 18$

Step 2 — Apply the Square Root Property:

$u = \pm\sqrt{18}$

Step 3 — Simplify the radical. The largest perfect square factor of $18$ is $9$ . Rewrite $18$ as $9 \cdot 2$ and apply the Product Property:

$\sqrt{18} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$

Therefore $u = \pm 3\sqrt{2}$ .

Rewrite to show two solutions: $u = 3\sqrt{2}$ or $u = -3\sqrt{2}$ .

Step 4 — Check both solutions by substituting into the original equation:

For $u = 3\sqrt{2}$ : $\frac{2}{3}(3\sqrt{2})^2 + 5 = \frac{2}{3}(9 \cdot 2) + 5 = \frac{2}{3}(18) + 5 = 12 + 5 = 17$ ✓

For $u = -3\sqrt{2}$ : $\frac{2}{3}(-3\sqrt{2})^2 + 5 = \frac{2}{3}(9 \cdot 2) + 5 = \frac{2}{3}(18) + 5 = 12 + 5 = 17$ ✓

The solutions are $u = 3\sqrt{2}$ and $u = -3\sqrt{2}$ . When the coefficient of the squared term is a fraction, multiplying both sides by the reciprocal is the most efficient way to obtain a coefficient of $1$ . This example also shows that the resulting constant may not be a perfect square, requiring simplification of the radical using the Product Property of Square Roots.

Updated 2026-04-21

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OpenStax Elementary Algebra

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