Question

Solve the equation. Remember to check for extraneous solutions. $$\frac{x}{6}+\frac{15}{x}=\frac{5}{6}$$

Short answer

The given equation does not have any real solutions.

Step-by-step solution

Clear the fractions

Multiply through by \(6x\) to eliminate the fractions. This yields \(x^2+90=5x\).

Rearrange into quadratic form

Rearrange the above equation to get it into a standard quadratic form \(ax^2+bx+c=0\). So, \(x^2-5x+90=0\).

Solve the quadratic equation

This equation can be solved either by factoring, completing the square, or using the quadratic formula. Here, the quadratic formula \((-b \pm \sqrt{b^2-4ac})/2a\) gives \(x = (5 \pm \sqrt{5^2-4*1*90})/ (2*1) = (5 \pm \sqrt{25-360})/2\). Since the root of the radicand \(25-360\) is negative and the square root of a negative number does not produce a real number, there are no real solutions.

Check for extraneous solutions

Since there are no potential solutions from Step 3, there is no need to check for extraneous solutions.