Question

Solve for all possible values of x.
square root of, 6, x, plus, 6, end square root, equals, x, plus, 1
6x+6
​
=
x+1

Answer 1

To solve the equation \(\sqrt{6x + 6} = x + 1\), let's follow these steps:

1. Square both sides to remove the square root:
\[
(\sqrt{6x + 6})^2 = (x + 1)^2
\]
This simplifies to:
\[
6x + 6 = (x + 1)(x + 1)
\]
\[
6x + 6 = x^2 + 2x + 1
\]

2. Rearrange the equation to set it to zero:
\[
0 = x^2 + 2x + 1 - 6x - 6
\]
\[
0 = x^2 - 4x - 5
\]

3. Factor the quadratic:
\[
0 = (x - 5)(x + 1)
\]

4. Set each factor to zero:
\[
x - 5 = 0 \implies x = 5
\]
\[
x + 1 = 0 \implies x = -1
\]

5. Check both solutions in the original equation \(\sqrt{6x + 6} = x + 1\):

- For \(x = 5\):
\[
\sqrt{6(5) + 6} = \sqrt{30 + 6} = \sqrt{36} = 6
\]
\[
5 + 1 = 6
\]
This is valid.

- For \(x = -1\):
\[
\sqrt{6(-1) + 6} = \sqrt{-6 + 6} = \sqrt{0} = 0
\]
\[
-1 + 1 = 0
\]
This is also valid.

Thus, the solutions to the equation \(\sqrt{6x + 6} = x + 1\) are:
\[
\boxed{5 \text{ and } -1}
\]