Question

Solve the equation. Check for extraneous solutions. $$x=\sqrt{-4 x-4}$$

Short answer

The given equation has no solutions as \(x = -2\) is an extraneous solution.

Step-by-step solution

Isolate the square root

The square root function can be isolated on one side by squaring both sides of the equation. Squaring the left side \(x\) gives \(x^2\), and squaring the square root on the right side will remove the square root, leaving us with \(-4x - 4\). Our new equation will be \(x^2 = -4x -4\)

Convert to quadratic equation

The equation \(x^2 = -4x - 4\) can be made into a standard quadratic equation by adding \(4x + 4\) to both sides of the equation. This step yields \(x^2 + 4x + 4 = 0\)

Factor the quadratic equation

The quadratic equation \(x^2 + 4x + 4 = 0\) can be factored as \((x + 2)^2 = 0\) or it could be solved by applying quadratic formula. However, factoring the quadratic equation is simpler in this case. After factoring, solve for \(x\) by setting \(x + 2 = 0\), which yields \(x = -2\)

Check for extraneous solutions

Substitute \(x = -2\) into the original equation to check if it is an extraneous solution. Substituting into the original equation yields \(-2 = \sqrt{-4*-2 - 4}\), which simplifies to \(-2 = \sqrt{4}\). This equation does not hold, meaning that \(x = -2\) is an extraneous solution and thus not a solution to the equation.