Question

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

no solution
no solution

x=5
x equals 5

x=3
x equals 3

x=3
and x=5

Answer 1

To solve the equation \( x + 8 = \sqrt{12x - 11} + 6 \), we'll first isolate the square root on one side:

1. Rearranging the equation gives us:
\[
\sqrt{12x - 11} = x + 8 - 6
\]
which simplifies to:
\[
\sqrt{12x - 11} = x + 2
\]

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

3. Expanding the right side gives:
\[
12x - 11 = x^2 + 4x + 4
\]

4. Rearranging the equation leads us to:
\[
0 = x^2 + 4x + 4 - 12x + 11
\]
Simplifying this gives us:
\[
0 = x^2 - 8x + 15
\]

5. We can factor this quadratic equation:
\[
0 = (x - 3)(x - 5)
\]

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

7. Now, we need to check both solutions in the original equation to ensure they are valid (as squaring can introduce extraneous solutions).

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

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

Since both \( x = 3 \) and \( x = 5 \) are valid solutions, the final answer is:
\[
\text{x = 3 and x = 5}
\]