solve the equation $ax^2-\sqrt(x)b-c=0$

Question

I need to solve this system:

\begin{align}y&=\sqrt x b+c\\ y&=ax^2\end{align}

where the $x$ and the $y$ are the variable.

Once I arrive to the equation:

$ax^2-\sqrt x b-c=0$

I don't know how to find $x$. Can you help me?

Answer 1

For $a=0$ it's obvious.

For $a\neq0$ after dividing by $a$ and assuming $\sqrt{x}=t$ we need to solve $$t^4+pt+q=0,$$ which we can solve by the Ferrari method: https://www.encyclopediaofmath.org/index.php/Ferrari_method

See also, here the beautiful explanation: https://www.youtube.com/watch?v=zefyAObryl8

Although, you can solve right the equation $x^2+x-3=0$. :)