Solve for x - square roots in equations

Question

How would I solve the following equation for x:

$ {\sqrt{33-24x}}=1+2{\sqrt{11-7x}} $

Is there even a solution, or am I missing something?

Answer 1

Graphical "evidence" (not a formal proof !) that there are no solutions:

Answer 2

We have

$$ ( \sqrt{ 33 - 24 x } - 2 \sqrt{ 11 - 7x} )^2 = 1 \iff (33-24x) - 4 \sqrt{ (33-24x)(11-7x)} + 4(11-7x) = 1 \iff -52 x + 76 = 4 \sqrt{ (33-24x)(11-7x)} \iff (76 - 52x)^2 = 16(33-24x)(11-7x) $$

And from here, you should be able to continue since it is a quadratic equation:

Added:

Notice we must have $33 - 24x \geq 0$ and $11 - 7x \geq 0$.