Solve equation $4 \sqrt{1-x} = x+6-3\sqrt{1-x^2}+5\sqrt{1+x}$

Question

I have some difficulties with this problem:

Solve the equation: $4 \sqrt{1-x} = x+6-3\sqrt{1-x^2}+5\sqrt{1+x}$

I tried to let's $\sqrt{1-x} = a$ and $\sqrt{1+x}=b$ then try to solve equations but it seems difficult.

Can anyone help me deal with this problem or recommend any idea? Thank you

Answer 1

Just so you know, your approach can be completed: $$4a + 3ab = b^2+5b+5\implies a^2(4+3b)^2 = (b^2+5b+5)^2$$ and $b^2+a^2 = 2$ and therefore: $$(2-b^2)(4+3b)^2 = (b^2+5b+5)^2.$$

If you use WA, then it factors as: $$(b+1)(5b+7)(2b^2+2b-1) = 0$$ and since $b = \sqrt{1+x}\geq 0,$ the only real solution you will get is: $$x = b^2-1 = \left(\dfrac{\sqrt{3}-1}{2}\right)^2-1 = -\dfrac{\sqrt{3}}{2}.$$

Answer 2

I multiplied the values out by hand and got $100x^4 -96x^3 -75x^2 + 72x$ with two visually obvious linear factors namely $x$ and $25x-24$ leaving $4x^2-3$. Easy to check only one root works.

What is the origin of this question? Anything interesting?