Questions: The solution set of the equation x-1=√(2x+6) is (1) 5,-1 (3) -1 (2) 5 (4)

Solution Steps

The given equation is:

\[ x - 1 = \sqrt{2x + 6} \]

To eliminate the square root, we first isolate it on one side of the equation, which is already done.

Square both sides of the equation to remove the square root:

\[ (x - 1)^2 = (\sqrt{2x + 6})^2 \]

This simplifies to:

\[ x^2 - 2x + 1 = 2x + 6 \]

Rearrange the equation to bring all terms to one side:

\[ x^2 - 2x + 1 - 2x - 6 = 0 \]

Simplify:

\[ x^2 - 4x - 5 = 0 \]

Factor the quadratic equation:

\[ (x - 5)(x + 1) = 0 \]

Set each factor equal to zero:

1. \( x - 5 = 0 \) gives \( x = 5 \)
2. \( x + 1 = 0 \) gives \( x = -1 \)

Substitute \( x = 5 \) back into the original equation:

\[ 5 - 1 = \sqrt{2(5) + 6} \] \[ 4 = \sqrt{10 + 6} \] \[ 4 = \sqrt{16} \] \[ 4 = 4 \]

This is true, so \( x = 5 \) is a valid solution.

Substitute \( x = -1 \) back into the original equation:

\[ -1 - 1 = \sqrt{2(-1) + 6} \] \[ -2 = \sqrt{-2 + 6} \] \[ -2 = \sqrt{4} \] \[ -2 = 2 \]

This is false, so \( x = -1 \) is not a valid solution.

Final Answer