Solve $\frac{7}{x+\sqrt{x+5}}+\frac{7}{x-\sqrt{x+5}}=8$

Question

Solve the equation: $$\dfrac{7}{x+\sqrt{x+5}}+\dfrac{7}{x-\sqrt{x+5}}=8.$$ I am not sure how to approach the problem. Should we first determine the domain? I think we can also check for every value we get for $x$ if the expression is defined, or not. The answers are $x=-\dfrac{5}{4}$ and $x=4.$ The numerators are equal and we also have resemblance in the denomitators. I am not sure how to use that. Thank you in advance! Happy holidays!

Answer 1

$$\dfrac{7}{x+\sqrt{x+5}}+\dfrac{7}{x-\sqrt{x+5}} = \frac{7(x-\sqrt{x+5})}{x^2-x-5} +\frac{7(x+\sqrt{x+5})}{x^2-x-5} \\ = \frac{14x}{x^2-x-5}$$ So the equation is equivalent to $$\frac{14x}{x^2-x-5} =8 \\ 14x =8x^2-8x-40 \\ 8x^2-22x-40=0$$

Answer 2

Note that:

$$\frac{7}{x+\sqrt{x+5}}+\frac{7}{x-\sqrt{x+5}}=\frac{14x}{x^2-x-5}$$

Answer 3

\begin{align*} \frac{7}{x + \sqrt{x+5}} + \frac{7}{x - \sqrt{x + 5}} = 8 & \Rightarrow \frac{14x}{x^{2} - x - 5} = 8\\\\ & \Rightarrow 8x^{2} - 22x - 40 = 0\\\\ & \Rightarrow 4x^{2} - 11x - 20 = 0\\\\ & \Rightarrow \left(x = - \frac{5}{4}\right)\vee(x = 4) \end{align*}

Substituting in the original equation, we conclude that they are solutions indeed.