We are given the equation:

\[
\sqrt{3x} = \sqrt{ax} + 2, \quad a \geq 0, \quad x > 0
\]

To solve for \( x \) in terms of \( a \), let's follow these steps:

### Step 1: Isolate one square root term
Start by isolating one of the square roots. We can subtract \( \sqrt{ax} \) from both sides:

\[
\sqrt{3x} - \sqrt{ax} = 2
\]

### Step 2: Eliminate the square roots
To eliminate the square roots, square both sides of the equation. This will give us:

\[
\left( \sqrt{3x} - \sqrt{ax} \right)^2 = 2^2
\]

Expanding both sides:

\[
(\sqrt{3x})^2 - 2 \cdot \sqrt{3x} \cdot \sqrt{ax} + (\sqrt{ax})^2 = 4
\]

Simplifying each term:

\[
3x - 2 \cdot \sqrt{3x \cdot ax} + ax = 4
\]

This simplifies further to:

\[
3x - 2 \sqrt{3a x^2} + ax = 4
\]

### Step 3: Express the square root as a simpler form
Notice that \( \sqrt{3a x^2} = \sqrt{3a} \cdot x \), so the equation becomes:

\[
3x - 2 \cdot \sqrt{3a} \cdot x + ax = 4
\]

Now, factor out \( x \) from the terms on the left:

\[
x(3 - 2\sqrt{3a} + a) = 4
\]

### Step 4: Solve for \( x \)
Now, solve for \( x \):

\[
x = \frac{4}{3 - 2\sqrt{3a} + a}
\]

### Conclusion
Thus, the solution for \( x \) in terms of \( a \) is:

\[
x = \frac{4}{3 - 2\sqrt{3a} + a}
\]

### Additional Details
Would you like further steps for verifying this solution, or any clarifications? Let me know if you'd like to explore more about this equation.

---

**Related Questions**:
1. How do we solve equations involving multiple square roots?
2. What is the impact of the parameter \( a \) on the solution?
3. How can we check if this solution is valid for a specific value of \( a \)?
4. How can we solve similar equations with higher powers or other roots?
5. What would happen if \( a = 0 \) in this equation?

**Tip**: Always isolate square roots first when solving equations involving them to simplify the steps.

Learn how to solve the equation √ 3x = √ax + 2 for x, where a ≥ 0 and x > 0. This detailed solution covers the steps to isolate square roots, eliminate them by squaring, and solve for x in terms of a. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation