Solution equation of type $\sqrt{a+x^2} + \sqrt{b+x^2}-c-d=0 \,\,\,\,\,\,\,\, x \in \mathbb{R}$

Question

I'm facing an equation of the type $$\sqrt{a+x^2} + \sqrt{b+x^2}-c-d=0 \,\,\,\,\,\,\,\, x \in \mathbb{R}$$ I'm not practical about tricks of resolution for equation, I'd just like to know the steps to reach the solutions: $$x_{\pm}= \pm \frac{1}{2(c+d)}\sqrt{[(c+d)^2-(a+b)^2][(c+d)^2-(a-b)^2]} $$

Answer 1

We start with $$\sqrt{a+x^2} + \sqrt{b+x^2}=c+d$$ squaring both parts to obtain $$2\sqrt{(a+x^2)(b+x^2)} + 2x^2+a+b=(c+d)^2$$ $$2\sqrt{(a+x^2)(b+x^2)} =- 2x^2-a-b+(c+d)^2$$ squaring again:$$4x^4+4x^2(a+b)+ab =4x^4-4(-a-b+(c+d)^2)x^2+((c+d)^2-a-b)^2$$ Finally, $$4x^2(c+d)^2=\left[(c+d)^2-a-b\right]^2-ab$$