1Cademy - Solving $$3\sqrt{3x-5}

Learn Before

How to Solve a Radical Equation

Example

Solving $3\sqrt{3x-5} - 8 = 4$

Solve the radical equation $3\sqrt{3x - 5} - 8 = 4$ using the four-step procedure. This equation has a numerical coefficient multiplying the square root, so isolating the radical means isolating the entire product $3\sqrt{3x - 5}$ — and when both sides are squared, the coefficient is squared along with the radical.

Step 1 — Isolate the radical. Add $8$ to both sides to move the constant:

$3\sqrt{3x - 5} = 12$

Step 2 — Square both sides. Because the left side is $3\sqrt{3x - 5}$ , squaring it means $(3\sqrt{3x - 5})^2 = 3^2 \cdot (\sqrt{3x - 5})^2 = 9(3x - 5)$ . On the right, $12^2 = 144$ :

$(3\sqrt{3x - 5})^2 = 12^2$

$9(3x - 5) = 144$

Step 3 — Solve the new equation. Distribute the $9$ :

$27x - 45 = 144$

Add $45$ to both sides: $27x = 189$ . Divide both sides by $27$ : $x = 7$ .

Step 4 — Check. Substitute $x = 7$ into the original equation:

$3\sqrt{3(7) - 5} - 8 = 3\sqrt{21 - 5} - 8 = 3\sqrt{16} - 8 = 3(4) - 8 = 12 - 8 = 4$

Since $4 = 4$ is true, $x = 7$ is confirmed as the solution. When a coefficient multiplies the radical, the coefficient must be squared along with the square root when applying the Squaring Property. Here, $(3\sqrt{3x-5})^2 = 9(3x-5)$ , not $3(3x-5)$ — forgetting to square the coefficient is a common error.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After