Question

Solve the equation. Remember to check for extraneous solutions. $$\frac{4}{x^{2}-2 x}=\frac{4}{3 x-6}$$

Short answer

The solution to the equation is x = 3

Step-by-step solution

Make the Denominator Same

First, recognize that both denominators can be simplified: \(x^{2}-2 x = x(x-2)\) and \(3x-6 = 3(x-2)\). So, rewrite the equation as: \(\frac{4}{x(x-2)} =\frac{4}{3(x-2)}\)

Cancel Out the Common Factors

Cancel out the common factors on both sides: \(4/x = 4/3\)

Cross Multiply and Solve for x

Cross multiply and solve the resulting equation for x: \(12 = 4x\) then \(x = 12/4 = 3\)

Check For Extraneous Solutions

Substitute x = 3 back into the original equation to check if it is valid: \(\frac{4}{(3)^{2}-2(3)} = \frac{4}{3(3)-6} \rightarrow \frac{4}{9-6} = \frac{4}{9-6} \rightarrow 2 = 2\). x = 3 is a valid solution, thus, there are no extraneous solutions