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The equation given is:

\[
\sqrt{\frac{2x}{5}} = \sqrt{3x - 58}
\]

To solve for \( x \), follow these steps:

### Step 1: Square both sides to eliminate the square roots.
\[
\left(\sqrt{\frac{2x}{5}}\right)^2 = \left(\sqrt{3x - 58}\right)^2
\]
This simplifies to:
\[
\frac{2x}{5} = 3x - 58
\]

### Step 2: Clear the fraction by multiplying all terms by 5.
\[
2x = 5(3x - 58)
\]

### Step 3: Distribute the 5 on the right side.
\[
2x = 15x - 290
\]

### Step 4: Move all terms involving \( x \) to one side of the equation.
\[
2x - 15x = -290
\]
\[
-13x = -290
\]

### Step 5: Solve for \( x \).
\[
x = \frac{-290}{-13} = \frac{290}{13} \approx 22.31
\]

Thus, the solution is \( x = \frac{290}{13} \) or approximately \( x = 22.31 \).

Would you like further details or have any other questions?

Here are 5 related questions that you might find useful:

1. What would happen if the equation had a cube root instead of a square root?
2. How would the solution change if the constants were different, e.g., replacing 58 with another number?
3. What are the possible restrictions on \( x \) for the original equation to be valid?
4. How can you check if the solution \( x = \frac{290}{13} \) satisfies the original equation?
5. What methods can be used to solve equations involving roots other than algebraic manipulation?

**Tip:** When solving equations involving square roots, always consider checking your solutions by substituting them back into the original equation to ensure they do not create any contradictions (like taking the square root of a negative number).

Solve the equation \(\sqrt{\frac{2x}{5}} = \sqrt{3x - 58}\).

Learn how to solve the radical equation \(\sqrt{\frac{2x}{5}} = \sqrt{3x - 58}\) with a step-by-step solution. This problem involves squaring both sides to eliminate square roots and applying algebraic techniques to solve for x. Suitable for students in Grades 9-11.

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