Question

In the following exercises, solve.
$15x^{-2}-4x^{-1}-4=0$

Short answer

The solutions of the given equation are $x=-\frac{5}{2}$ and $x=\frac{3}{2}$.

Step-by-step solution

Step 1. Write the equation in standard form.

The given equation is:

$15x^{-2}-4x^{-1}-4=0$

It can be written as:

$15(x^{-1}{)}^2-4x^{-1}-4=0$

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

role="math" localid="1644992521952" $15u^2-4u-4=0$

Step 2. Solve the equation.

Split the middle term.

$\begin{array}{rcl}15u^2-10u+6u-4& =& 0\\ 5u(3u-2)+2(3u-2)& =& 0\\ (5u+2)(3u-2)& =& 0\\ u& =& -\frac{2}{5}\text{and}u=\frac{2}{3}\end{array}$

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

$\begin{array}{rcl}{x}^{-1}& =& -\frac{2}{5}\text{and}{x}^{-1}=\frac{2}{3}\\ x& =& -\frac{5}{2}\text{and}x=\frac{3}{2}\end{array}$

Step 3. Conclusion.

The solutions of the given equation are$-\frac{5}{2}$ and $x=\frac{3}{2}$.