<s>[INST] How to solve this kind of equation? I have an equation (in my homework) of the form
$a=\sqrt{x^2 + b^2} + \sqrt{x^2 + c^2}$
which I would like to solve for $x$.  I am not sure how best to proceed.  My thought is to square both sides of the equation, which gives me
$a^2=b^2+c^2+2x^2+\sqrt{(b^2+x^2)\times(c^2+x^2)}$
and then
$a^2=b^2+c^2+2x^2+\sqrt{b^2c^2 + b^2x^2 + c^2x^2 + x^4}$
but I am not sure what to do with the square root in order to eventually solve for $x$.  How do I finish solving this, or am going the wrong way?
 [/INST] Suppose that our equation 
$$\sqrt{x^2+b^2}+\sqrt{x^2+c^2}=a.\tag{1}$$
holds. Assume $a\ne 0$ and $b\ne c$. Flip both sides over. We get 
$$\frac{1}{\sqrt{x^2+b^2}+\sqrt{x^2+c^2}}=\frac{1}{a}.$$
Multiply top and bottom on the left by $\sqrt{x^2+b^2}-\sqrt{x^2+c^2}$. We get
$$\frac{\sqrt{x^2+b^2}-\sqrt{x^2+c^2}}{b^2-c^2}=\frac{1}{a}.$$
Conclude that 
$$\sqrt{x^2+b^2}-\sqrt{x^2+c^2}=\frac{b^2-c^2}{a}.\tag{2}$$
Add (1) and (2).   Now we know $2\sqrt{x^2+b^2}$, and the rest is downhill.
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