How should I go around on proving the quadratic equation $$a^2 x^2 +(2ac-b^2)x+c^2=0$$ having real and positive solutions?
I tried to use the fact that if a quadratic equation has real and positive solutions, then the discriminant is greater or equal to 0, and that $\frac{b}{a}<0$ and $\frac{c}{a}>0$. But I kind of stuck after proving that $\frac{c^2}{a^2}>0$ and that $\frac{c}{a}>0$.
Hint:
This is a standard high school exercise. You indeed have to prove that $$\Delta=(2ac-b^2)^2-4a^2c^2=b^2(b^2-4ac)$$ is positive for the existence of real roots, that the product of these roots is positive (nonzero roots with the same sign) and their sum is also positive (the common sign is also the sign of the sum).