Paul's Online Notes
Home / Algebra / Solving Equations and Inequalities / Equations Reducible to Quadratic in Form
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 2.9 : Equations Reducible to Quadratic in Form

2. Solve the following equation.

${x^{ - 4}} - 7{x^{ - 2}} - 18 = 0$

Show All Steps Hide All Steps

Hint : Remember to look at the exponents of the first two terms and try to find a substitution that will turn this into a “normal” quadratic equation.
Start Solution

First let’s notice that $$- 4 = 2\left( { - 2} \right)$$ and so we can use the following substitution to reduce the equation to a quadratic equation.

$u = {x^{ - 2}}\hspace{0.25in}\hspace{0.25in}{u^2} = {\left( {{x^{ - 2}}} \right)^2} = {x^{ - 4}}$ Show Step 2

Using this substitution the equation becomes,

\begin{align*}{u^2} - 7u - 18 & = 0\\ \left( {u - 9} \right)\left( {u + 2} \right) & = 0\end{align*}

We can easily see that the solution to this equation is : $$u = - 2$$ and $$u = 9$$ .

Show Step 3

Now all we need to do is use our substitution from the first step to determine the solution to the original equation.

$u = - 2:\hspace{0.25in}{x^{ - 2}} = \frac{1}{{{x^2}}} = - 2\hspace{0.25in} \Rightarrow \hspace{0.25in}{x^2} = - \frac{1}{2}\hspace{0.25in} \Rightarrow \hspace{0.25in}x = \pm \sqrt { - \frac{1}{2}} = \pm \frac{1}{{\sqrt 2 }}i$ $u = 9:\hspace{0.25in}{x^{ - 2}} = \frac{1}{{{x^2}}} = 9\hspace{0.25in} \Rightarrow \hspace{0.25in}{x^2} = \frac{1}{9}\hspace{0.25in} \Rightarrow \hspace{0.25in}x = \pm \sqrt {\frac{1}{9}} = \pm \frac{1}{3}$

Therefore the four solutions to the original equation are : $\require{bbox} \bbox[2pt,border:1px solid black]{{x = \pm \frac{1}{{\sqrt 2 }}i\,\,{\mbox{and }}x = \pm \frac{1}{3}}}$ .