Question

Solve the radical equation. Check all proposed solutions.
x minus StartRoot 3 x minus 2 EndRoot equals 4

Answer 1

To solve the equation \( x - \sqrt{3x - 2} = 4 \), let's start by isolating the square root term.

1. Rearranging the equation gives:
\[
\sqrt{3x - 2} = x - 4
\]

2. Now, square both sides to eliminate the square root:
\[
(\sqrt{3x - 2})^2 = (x - 4)^2
\]
This simplifies to:
\[
3x - 2 = (x - 4)(x - 4)
\]
Expanding the right side:
\[
3x - 2 = x^2 - 8x + 16
\]

3. Rearranging the equation to bring everything to one side yields:
\[
0 = x^2 - 8x + 16 - 3x + 2
\]
Simplifying gives:
\[
0 = x^2 - 11x + 18
\]
or
\[
x^2 - 11x + 18 = 0
\]

4. Now, we can factor the quadratic:
\[
(x - 2)(x - 9) = 0
\]
Therefore, the solutions are:
\[
x = 2 \quad \text{and} \quad x = 9
\]

5. Now we will check both proposed solutions to ensure they do not introduce extraneous solutions:

- **Checking \( x = 2 \)**:
\[
2 - \sqrt{3(2) - 2} = 2 - \sqrt{6 - 2} = 2 - \sqrt{4} = 2 - 2 = 0 \quad \text{(not satisfied)}
\]

- **Checking \( x = 9 \)**:
\[
9 - \sqrt{3(9) - 2} = 9 - \sqrt{27 - 2} = 9 - \sqrt{25} = 9 - 5 = 4 \quad \text{(satisfied)}
\]

Thus, after verifying both potential solutions, the only valid solution to the original equation is:
\[
\boxed{9}
\]