<s>[INST] Solve for $x$ in $\sqrt{3.5^2+x^2}-\sqrt{3.0^2+x^2}=0.25$ I am taking physics right now and I have gotten my problem down to the following equation:
$$\sqrt{3.5^2+x^2}-\sqrt{3.0^2+x^2}=0.25$$
I am looking for some guidance as to what to do with the square roots in order to solve for $x$. I know I can't take the square of both sides and I tried to factor and failed.
 [/INST] The following is a shortcut that works for this particular equation.
Multiplying both sides by the sum of the radicals (which is always $\ne 0$) gives:
$$(3.5^2+x^2)-(3.0^2+x^2)=0.25(\sqrt{3.5^2+x^2}+\sqrt{3.0^2+x^2})$$
$$\sqrt{3.5^2+x^2}+\sqrt{3.0^2+x^2} = 13$$
Adding the latter to the original equation gives:
$$2 \sqrt{3.5^2+x^2} = 13.25$$
From there $x^2 = (\frac{13.25}{2})^2 - 3.5^2 = 43.890625 - 12.25 = 31.640625$, thus $x = \pm 5.625$.
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