Question

Solve the equation. Check for extraneous solutions. $$2 x=\sqrt{-13 x-10}$$

Short answer

The equation \(2x = \sqrt{-13x - 10}\) has no valid solutions since both potential solutions \(-\frac{5}{4}\) and \(-2\) are extraneous.

Step-by-step solution

Squaring both sides of the equation

To eliminate the square root, square both sides of the equation to get a quadratic equation. So \((2x)^2 = (\sqrt{-13x-10})^2\) which simplifies to \(4x^2 = -13x - 10\).

Solving the quadratic equation

Rearrange terms and set the equation to zero: \(4x^2 + 13x + 10 = 0\). Factoring this quadratic equation yields \((4x+5)(x+2) = 0\). Setting each factor equal to zero gives two potential solutions: \(x=-\frac{5}{4}\) and \(x=-2\)

Checking for extraneous solutions

Substitute these potential solutions back into the original equation to verify if they are valid. Substituting \(x=-\frac{5}{4}\) comes out false as \(-\frac{5}{2} \neq \sqrt{-13(-\frac{5}{4})-10}\). Substituting \(x=-2\) gives us \(-4=\sqrt{-13(-2)-10}\) which is also false. After squaring both sides, it comes out to be 16 equals to 36, which is false. Therefore, both solutions are extraneous, and the original equation has no solution.