I tried to solving this it became 4th power of equation

Question

Kindly see this image and solve it.

I solved this. But it form fourth power of equation. It is a 10th grade math. It seems any other techniq ond logic will apply please solve it. I didn't sleep whole night for this question

Answer 1

I am going to prove that the only real solution of the equation:

$\sqrt\frac{x}{1-x}+\sqrt{1-x}=1$

is $x=0$.

Proof:

It is necessary that $x\in\left[0,1\right[$ in order that the values inside the two square roots are nonnegative.

Squaring both sides of the equation, we obtain:

$\frac{x}{1-x}+1-x+2\sqrt{x}=1$

$\frac{x}{1-x}-x+2\sqrt{x}=0$

$\frac{x-x+x^2}{1-x}+2\sqrt{x}=0$

$\frac{x^2}{1-x}+2\sqrt{x}=0$

But $\frac{x^2}{1-x}\ge0$ and $2\sqrt{x}\ge0$, so there is only a possibility which is:

$\frac{x^2}{1-x}=2\sqrt{x}=0$

Therefore $x=0$.

As a matter of fact $x=0$ is a real solution of the equation and it is actually the only one.