Question

In the following exercises, solve.
$8x^{-2}-2x^{-1}-3=0$

Short answer

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

Step-by-step solution

Step 1. Write the equation in standard form.

The given equation is:

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

It can be written as:

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

Substitute $x^{-1}=u$.

$8u^2-2u-3=0$

Step 2. Solve the equation.

Split the middle term.

$\begin{array}{rcl}8u^2-6u+4u-3& =& 0\\ 2u(4u-3)+1(4u-3)& =& 0\\ (2u+1)(4u-3)& =& 0\\ u& =& -\frac{1}{2}\text{and}u=\frac{3}{4}\end{array}$

Substitute $u=x^{-1}$.

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

Step 3. Conclusion.

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