Question

Solve the equation. Check for extraneous solutions. $$x=\sqrt{\frac{3}{2} x+\frac{5}{2}}$$

Short answer

The solution to the equation is x = 1.

Step-by-step solution

Isolate the Square Root

First, work on isolating the square root from the variable x. The equation is already isolated in this formulation, so we can move to the next step.

Squaring Both Sides

Now, eliminate the square root by squaring both sides using the property \((a = b \Rightarrow a^2 = b^2)\). This results in \(x^2 = \frac{3}{2}x + \frac{5}{2}\)

Rearrange and Simplify

Rearrange the equation so that all terms are on one side: \(x^2 - \frac{3}{2}x - \frac{5}{2} = 0\)

Solve for x

Solve the quadratic equation. The roots are obtained using the quadratic formula \((x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})\). The solutions for x are: \(x = 1\) and \(x = -\frac{5}{2}\)

Check for Extraneous Solutions

Substitute both solutions back into the original equation to verify if they satisfy it. The solution \(x = 1\) satisfies the equation, however the solution \(x = -\frac{5}{2}\) does not satisfy the original equation. So, \(x = -\frac{5}{2}\) is an extraneous solution.