<s>[INST] What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}, x \in \mathbb{R}$ if $\sqrt{49-x^2}-\sqrt{25-x^2}=3$ Suppose that real number $x$ satisfies $$\sqrt{49-x^2}-\sqrt{25-x^2}=3$$What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$?

This is what I did:
I try to multiply by the conjugate. Its value I believe is technically the solution. $(\sqrt {49-x^2} + \sqrt {25-x^2})(\sqrt {49-x^2} - \sqrt {25-x^2})=24$. Given that $(\sqrt {49-x^2} - \sqrt {25-x^2}) = 3$, $(\sqrt {49-x^2} + \sqrt {25-x^2})(\sqrt {49-x^2} - \sqrt {25-x^2})= 3(\sqrt {49-x^2} + \sqrt {25-x^2}) =24\implies \sqrt {49-x^2} + \sqrt {25-x^2}=8$ 

My question is that will this method work for similar problems, and is there a faster method?
Thanks!
 [/INST] You basically used the formula $$a+b=\frac{a^2-b^2}{a-b}$$ 
in an indirect way.
The problem gave you $a-b$ and, since $a,b$ are square roots of simple expressions, it is easy to calculate $a^2-b^2$.
And for the same reason, the same trick can be used to calculate 
$$\sqrt{\mbox{nice}}\pm \sqrt{\mbox{nice}}$$
whenever when $\sqrt{\mbox{nice}}\mp \sqrt{\mbox{nice}}$ is given.
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