Question

Solve the equation. Check for extraneous solutions. $$x=\sqrt{6 x-9}$$

Short answer

The solution to the equation is \(x = 3\).

Step-by-step solution

Square both sides

In this step, square both sides of the equation to eliminate the square root. The equation becomes: \(x^2 = (6x-9)\}. Now simplify the right hand side to get \(x^2 - 6x + 9 = 0\)

Solve the quadratic equation

The equation in step 1 is a quadratic equation. Solve it by factorising, thereby obtaining two potential solutions for \(x\). For \(x = 3\), check them against the original equation in order to confirm whether they are valid solutions.

Check for extraneous solutions

After calculating the roots as \(x = 3\), substitute them back into the original equation and verify whether the left-hand side equals the right-hand side. For \(x = 3\), it does, so there are no extraneous solutions in this case.