Example

Solving 5n49=0\sqrt{5n-4} - 9 = 0

Solve the radical equation 5n49=0\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 9-9 term. Add 99 to both sides to move it: 5n49+9=0+9\sqrt{5n - 4} - 9 + 9 = 0 + 9 5n4=9\sqrt{5n - 4} = 9

Step 2 — Square both sides. Apply the Squaring Property to remove the radical: (5n4)2=92(\sqrt{5n - 4})^2 = 9^2 5n4=815n - 4 = 81

Step 3 — Solve the new equation. Add 44 to both sides: 5n=855n = 85 Divide both sides by 55: n=17n = 17

Step 4 — Check. Substitute n=17n = 17 into the original equation: 5(17)49=8549=819=99=0\sqrt{5(17) - 4} - 9 = \sqrt{85 - 4} - 9 = \sqrt{81} - 9 = 9 - 9 = 0 Since 0=00 = 0 is true, n=17n = 17 is confirmed as the solution.

Unlike simpler examples where the radical is already isolated, this equation requires an algebraic step (adding 99 to both sides) before the squaring step can be applied.

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Updated 2026-06-25

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