Solve the radical equation 5x+5−−−−−√−x=1

Question

.(1 point)
Responses

x=1
x equals 1

x=4
x equals 4

x=4
and x=−1
x equals 4 and x equals negative 1

x=1
and x=4

Answer 1

To solve the radical equation \( \sqrt{5x + 5} - x = 1 \), we will first isolate the square root and then square both sides.

Here are the steps:

1. Isolate the square root:

\[ \sqrt{5x + 5} = x + 1 \]

2. Square both sides:

\[ 5x + 5 = (x + 1)^2 \]

3. Expand the right side:

\[ 5x + 5 = x^2 + 2x + 1 \]

4. Rearrange the equation:

\[ 0 = x^2 + 2x + 1 - 5x - 5 \]

\[ 0 = x^2 - 3x - 4 \]

5. Factor the quadratic:

\[ 0 = (x - 4)(x + 1) \]

6. Set each factor to zero:

\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

So, the potential solutions are \( x = 4 \) and \( x = -1 \).

7. Verify both solutions in the original equation:

• For \( x = 4 \): \[ \sqrt{5(4) + 5} - 4 = \sqrt{20 + 5} - 4 = \sqrt{25} - 4 = 5 - 4 = 1 \quad \text{(valid)} \]

• For \( x = -1 \): \[ \sqrt{5(-1) + 5} - (-1) = \sqrt{-5 + 5} + 1 = \sqrt{0} + 1 = 0 + 1 = 1 \quad \text{(valid)} \]

Both solutions satisfy the original equation.

Thus, the solutions are \( x = 4 \) and \( x = -1 \).

The correct response is: x equals 4 and x equals negative 1.