1Cademy - Solving $$\sqrt{5n-4}

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How to Solve a Radical Equation

Example

Solving $\sqrt{5n-4} - 9 = 0$

Solve the radical equation $\sqrt{5n - 4} - 9 = 0$ by applying the four-step procedure for radical equations.

Step 1 — Isolate the radical. The square root is not alone on the left side because of the $-9$ term. Add $9$ to both sides to move it:

$\sqrt{5n - 4} - 9 + 9 = 0 + 9$

$\sqrt{5n - 4} = 9$

Step 2 — Square both sides. Apply the Squaring Property to remove the radical:

$(\sqrt{5n - 4})^2 = 9^2$

$5n - 4 = 81$

Step 3 — Solve the new equation. Add $4$ to both sides: $5n = 85$ . Divide both sides by $5$ : $n = 17$ .

Step 4 — Check. Substitute $n = 17$ into the original equation:

$\sqrt{5(17) - 4} - 9 = \sqrt{85 - 4} - 9 = \sqrt{81} - 9 = 9 - 9 = 0$

Since $0 = 0$ is true, $n = 17$ is confirmed as the solution. Unlike the simpler example $\sqrt{2x - 1} = 7$ where the radical is already isolated, this equation requires an algebraic step (adding $9$ to both sides) before the squaring step can be applied.

Updated 2026-04-21

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OpenStax Elementary Algebra

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