Question

Solve the given equation.

$8x^{-2}-2x^{-1}-3=0$

Short answer

The solutions are$x=-2$and$x=\frac{4}{3}$.

Step-by-step solution

Step 1. Given information.

The given equation is$8x^{-2}-2x^{-1}-3=0$.

Step 2. Simplify the equation.

$\begin{array}{rcl}8x^{-2}-2x^{-1}-3& =& 0\\ 8x^{-2}-6x^{-1}+4x^{-1}-3& =& 0\\ 2x^{-1}\left(4x^{-1}-3\right)+1\left(4x^{-1}-3\right)& =& 0\\ \left(4x^{-1}-3\right)\left(2x^{-1}+1\right)& =& 0\end{array}$

Step 3. Now solve the equation for x.

$\begin{array}{rcl}4x^{-1}-3& =& 0\\ 4x^{-1}& =& 3\\ x^{-1}& =& \frac{3}{4}\\ \frac{1}{x}& =& \frac{3}{4}\\ x& =& \frac{4}{3}\end{array}$ and $\begin{array}{rcl}2x^{-1}+1& =& 0\\ 2x^{-1}& =& -1\\ x^{-1}& =& \frac{-1}{2}\\ \frac{1}{x}& =& \frac{-1}{2}\\ x& =& -2\end{array}$

Step 4. Simplified answer.

Hence, the required solutions are$x=-2$and$x=\frac{4}{3}$.