1Cademy - Solving $$\sqrt{4z-3} = \sqrt{3z+2}$$

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How to Solve a Radical Equation

Example

Solving $\sqrt{4z-3} = \sqrt{3z+2}$

Solve the radical equation $\sqrt{4z - 3} = \sqrt{3z + 2}$ , which has square root expressions on both sides of the equation.

Step 1 — Isolate the radical. Both radical terms are already isolated — each side of the equation consists of a single square root expression, so no rearrangement is needed.

Step 2 — Square both sides. Apply the Squaring Property to each side independently:

$(\sqrt{4z - 3})^2 = (\sqrt{3z + 2})^2$

$4z - 3 = 3z + 2$

Step 3 — Solve the new equation. Both radicals have been eliminated, leaving a linear equation. Subtract $3z$ from both sides: $z - 3 = 2$ . Add $3$ to both sides: $z = 5$ .

Step 4 — Check. Substitute $z = 5$ into the original equation:

$\sqrt{4(5) - 3} = \sqrt{3(5) + 2}$

$\sqrt{17} = \sqrt{17}$

Since $\sqrt{17} = \sqrt{17}$ is true, $z = 5$ is confirmed as the solution. When a radical equation has square roots on both sides, squaring eliminates both radicals simultaneously — the Squaring Property is applied to each side, and the result is a standard linear equation with no radicals remaining.

Updated 2026-04-21

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OpenStax Elementary Algebra

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