Ask Uncle Colin: An Unknown Denominator

Dear Uncle Colin,

I need to solve $\frac{1}{(x-10{)}^2}=4$ and all the explanations I’ve seen online are much more difficult than anything we’ve done in class. How would you approach it?

- Fractions Are Rightfully Execrated, Yes?

Hi, FAREY, and thanks for your message!

I think the simplest thing to do here is to rearrange:

• $1=4(x-10{)}^2$
• $\frac{1}{4}=(x-10{)}^2$

The square roots of $\frac{1}{4}$ are $\pm \frac{1}{2}$, so we need to solve $x-10=\frac{1}{2}$ and $x-10=-\frac{1}{2}$.

Those give the solutions of $x=\frac{21}{2}$ and $x=\frac{19}{2}$. You might prefer to write those as $10.5$ and $9.5$, but those look uglier to me.

Hope that helps!

- Uncle Colin