Question

Solve the equation. Remember to check for extraneous solutions. $$\frac{3}{x}=\frac{x}{12}$$

Short answer

The solutions to the equation are x = 6 and x = -6. Both are valid solutions, so there are no extraneous solutions.

Step-by-step solution

Remove the Fractions

To remove the fractions, cross-multiply the equation to get '3*12 =x*x', which simplifies to '36 =x^2'.

Solve the Resulting Equation

Take the square root of both sides of the equation to solve for x. Because both positive and negative roots are possible, we have two solutions: x = 6 and x = -6.

Check for Extraneous Solutions

Substitute both solutions back into the original equation to check that they work. For x=6, the left hand side becomes \( \frac{3}{6} \) and the right hand side becomes \( \frac{6}{12} \). Both simplify to 0.5, which means x=6 is a valid solution. For x=-6, the left hand side is \( \frac{3}{-6} \) and the right side becomes \( \frac{-6}{12} \). Both simplify to -0.5, indicating that x=-6 is also a valid solution. Thus, there are no extraneous solutions.