trying to solve a quadratic equation with surdic term.

Question

I am trying to come up with a solution for this quadratic equation, I've not gotten any meaningful solutions. Please help:

$$y^2–11\sqrt{y}+24 = 0$$

Answer 1

substituting $$\sqrt{y}=t$$ then you have to solve $$t^4-11t+24=0$$

Answer 2

Looks like a typo. The equation should be y^2-11y+24 = 0. you can then find the prime factors of 24 to come up with 8 and 3 as the roots of the equation.

Answer 3

By AM-GM $$y^2-11\sqrt{y}+24=y^2+8+8+8-11\sqrt{y}\geq4\sqrt[4]{8^3y^2}-11\sqrt{y}=\left(4\sqrt[4]{512}-11\right)\sqrt{y}>0,$$ which says that our equation has no real roots.

If you mean to solve $$y-11\sqrt{y}+24=0$$ then it's $$(\sqrt{y}-3)(\sqrt{y}-8)=0,$$ which gives $$y\in\{9,64\}.$$