How to solve an equation with a root in the divisor

Question

The given function is $$f(x)=\frac{4-x^2}{3-\sqrt{x^2+5}}-6$$

Solve for $f(x) = 0$ What's x?

Answer 1

You first have to determine the domain of validity of this equation: we must have $$3-\sqrt{x^2+5}\ne 0\iff x^2+5\ne 9\iff x\ne \pm 4.$$ \begin{align} \text{Now }\qquad f(x)=0&\iff\frac{4-x^2}{3-\sqrt{x^2+5}}=6\iff 4-x^2=18-6\sqrt{x^2+5}\\ &\iff x^2+14= 6\sqrt{x^2+5}\iff x^4 +28x^2+196=36x^2+180\\ &\iff x^4 -8x^2+16=(x^2-4)^2=0\qquad \end{align} Conclusion?

Answer 2

HINT For $z\ne 3$, $$ \frac{4-x^2}{3-\sqrt{z}} - 6 = 0 \iff 4-x^2 = 6(3-\sqrt{z}) $$ can you solve for $\sqrt{z}$ and square both sides?

Answer 3

Hints: Clear denominators, isolate the square root, square everything and hope for the best.