Question

Identify the solution of the square root equation x+8=12x−11−−−−−−−√+6.(1 point)
Responses

x=3
x equals 3

x=3 and x=5
x equals 3 and x equals 5

no solution
no solution

x=5

Answer 1

To solve the equation \( x + 8 = \sqrt{12x - 11} + 6 \), we start by isolating the square root term.

1. Rearranging the equation yields:
\[
\sqrt{12x - 11} = x + 8 - 6
\]
\[
\sqrt{12x - 11} = x + 2
\]

2. Next, we square both sides to eliminate the square root:
\[
(\sqrt{12x - 11})^2 = (x + 2)^2
\]
\[
12x - 11 = x^2 + 4x + 4
\]

3. Rearranging gives:
\[
0 = x^2 + 4x + 4 - 12x + 11
\]
\[
0 = x^2 - 8x + 15
\]

4. This is a quadratic equation. We can factor it:
\[
0 = (x - 3)(x - 5)
\]

5. Setting each factor equal to 0 gives the potential solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

6. We need to check both solutions in the original equation to see if they are valid.

- For \( x = 3 \):
\[
3 + 8 = \sqrt{12(3) - 11} + 6
\]
\[
11 = \sqrt{36 - 11} + 6
\]
\[
11 = \sqrt{25} + 6
\]
\[
11 = 5 + 6 \quad \text{(true)}
\]

- For \( x = 5 \):
\[
5 + 8 = \sqrt{12(5) - 11} + 6
\]
\[
13 = \sqrt{60 - 11} + 6
\]
\[
13 = \sqrt{49} + 6
\]
\[
13 = 7 + 6 \quad \text{(true)}
\]

Both solutions are valid. Therefore, the solution to the equation is:
\[
x = 3 \text{ and } x = 5
\]

So the correct response is: \( x = 3 \text{ and } x = 5 \).