Question

Solve the square root equation x−6=x−6−−−−√.(1 point)
Responses

x=6 and x=7
x equals 6 and x equals 7

no solution
no solution

x=7
x equals 7

x=6

Answer 1

To solve the equation \( x - 6 = \sqrt{x - 6} \), let's start by letting \( y = x - 6 \). Then the equation becomes:

\[
y = \sqrt{y}
\]

Next, we will square both sides to eliminate the square root:

\[
y^2 = y
\]

Rearranging this gives us:

\[
y^2 - y = 0
\]

Factoring out \( y \):

\[
y(y - 1) = 0
\]

This yields two solutions:

\[
y = 0 \quad \text{or} \quad y = 1
\]

Now, recall that \( y = x - 6 \):

1. If \( y = 0 \):
\[
x - 6 = 0 \implies x = 6
\]

2. If \( y = 1 \):
\[
x - 6 = 1 \implies x = 7
\]

Thus, the solutions to the equation \( x - 6 = \sqrt{x - 6} \) are \( x = 6 \) and \( x = 7 \).

**Final Answers:**
\( x = 6 \) and \( x = 7 \)

So the correct response is:

**x = 6 and x = 7**