Question

Solve the radical equation 3x−11−−−−−−√+5=x+2.(1 point)
Responses

x=4 or x=−5
x equals 4 or x equals negative 5

x=4 or x=5
x equals 4 or x equals 5

x=−5
x equals negative 5

x=4

Answer 1

To solve the equation \( \sqrt{3x - 11} + 5 = x + 2 \), we can start by isolating the radical term.

1. Subtract 5 from both sides:
\[
\sqrt{3x - 11} = x + 2 - 5
\]
Simplifying gives:
\[
\sqrt{3x - 11} = x - 3
\]

2. Next, square both sides to eliminate the square root:
\[
3x - 11 = (x - 3)^2
\]

3. Expand the right side:
\[
3x - 11 = x^2 - 6x + 9
\]

4. Rearrange the equation:
\[
0 = x^2 - 6x + 9 - 3x + 11
\]
Combine like terms:
\[
0 = x^2 - 9x + 20
\]

5. This can be rewritten as:
\[
x^2 - 9x + 20 = 0
\]

6. Now, factor the quadratic:
\[
(x - 4)(x - 5) = 0
\]

7. Set each factor to zero:
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

8. We have potential solutions \( x = 4 \) and \( x = 5 \). We should check for extraneous solutions by substituting back into the original equation.

- **Checking \( x = 4 \)**:
\[
\sqrt{3(4) - 11} + 5 = 4 + 2
\]
\[
\sqrt{12 - 11} + 5 = 6
\]
\[
1 + 5 = 6 \quad \text{(True)}
\]

- **Checking \( x = 5 \)**:
\[
\sqrt{3(5) - 11} + 5 = 5 + 2
\]
\[
\sqrt{15 - 11} + 5 = 7
\]
\[
\sqrt{4} + 5 = 7
\]
\[
2 + 5 = 7 \quad \text{(True)}
\]

Both solutions \( x = 4 \) and \( x = 5 \) are valid.

Thus, the final answer is:
**x = 4 or x = 5**.