Solve the radical equation 3x−11−−−−−−√+5=x+2

Question

3
𝑥
−
11
+
5
=
𝑥
+
2
.(1 point)
Responses

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

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

x=−5
𝑥
=
−
5
x equals negative 5

x=4

Answer 1

To solve the equation \(\sqrt{3x - 11} + 5 = x + 2\), we can follow these steps:

1. Isolate the square root: \[ \sqrt{3x - 11} = x + 2 - 5 \] \[ \sqrt{3x - 11} = x - 3 \]

2. Square both sides: \[ 3x - 11 = (x - 3)^2 \] \[ 3x - 11 = x^2 - 6x + 9 \]

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

4. Factor the quadratic: \[ 0 = (x - 4)(x - 5) \]

5. Find the solutions: \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

6. Check for extraneous solutions:

   • Plug \(x = 4\) back into the original equation: \[ \sqrt{3(4) - 11} + 5 = 4 + 2 \] \[ \sqrt{12 - 11} + 5 = 6 \] \[ 1 + 5 = 6 \quad \text{(true)} \]

   • Plug \(x = 5\) back into the original equation: \[ \sqrt{3(5) - 11} + 5 = 5 + 2 \] \[ \sqrt{15 - 11} + 5 = 7 \] \[ \sqrt{4} + 5 = 7 \] \[ 2 + 5 = 7 \quad \text{(true)} \]

Both solutions are valid.

Thus, the solutions to the equation are: \[ x = 4 \quad \text{and} \quad x = 5 \]

So the correct answer is: x=4 or x=5.