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