Solve each equation. √(3√(2x+3)) = √(5x-6)

Verified step by step guidance

1. Start by writing down the given equation: \(\sqrt{3\sqrt{2x+3}} = \sqrt{5x - 6}\).
2. Since both sides are square roots, square both sides of the equation to eliminate the outer square roots: \(\left(\sqrt{3\sqrt{2x+3}}\right)^2 = \left(\sqrt{5x - 6}\right)^2\).
3. Simplify both sides after squaring: \$3\sqrt{2x+3} = 5x - 6$.
4. Isolate the remaining square root term: \(\sqrt{2x+3} = \frac{5x - 6}{3}\).
5. Square both sides again to remove the square root: \(\left(\sqrt{2x+3}\right)^2 = \left(\frac{5x - 6}{3}\right)^2\), which simplifies to \$2x + 3 = \frac{(5x - 6)^2}{9}$.

Key Concepts

Properties of Square Roots

Understanding how to manipulate square roots is essential, including the fact that √a = √b implies a = b when both sides are nonnegative. This allows us to set the expressions inside the square roots equal to each other to solve the equation.

Nested Radicals

The equation involves a nested radical, √(3√(2x+3)), which requires careful simplification. Recognizing how to simplify or rewrite nested square roots helps in isolating the variable and solving the equation.

Solving Radical Equations and Checking for Extraneous Solutions

When solving equations involving radicals, squaring both sides can introduce extraneous solutions. It is important to check all solutions in the original equation to ensure they are valid and satisfy domain restrictions.