Is $x^{\frac{1}{2}}+ 2x+3=0$ a quadratic equation

Solution 1:

Yes, but the quadratic is not in x.

$ 2 (\sqrt x )^2 + (\sqrt x ) + 3 = 0$ can be considered to be quadratic equation in $(\sqrt x ).$

Solution 2:

That would not be considered quadratic in $x$, but you can let $u=\sqrt{x}$ to get $2u^2+u+3$. It WOULD be quadratic in $u$.

Solution 3:

if you substitute $\sqrt{x}=u$
so $$ \sqrt{x}+2x+3=u+2u^2+3$$

Solution 4:

Equations need equal signs, just like sentences need verbs. Just as the equation $ax^2 + bx + c = 0$ is quadratic in $x$, the equation $x^{1/2} + 2x + 3 = 0$ is quadratic in $x^{1/2}$ since it can be written in the form $2(x^{1/2})^2 + x^{1/2} + 3 = 0.$