Question

Find the smallest possible value of a quadratic equation

Answer 1

I'm assuming that you mean to find the minimum of a parabola, i.e. the minimum of a function defined as

$f(x) = ax^2+bx+c,\quad a,b,c \in\mathbb{R},\quad a\neq 0$

To find the minimum of a function, we have to find a point $x_0$ such that

$f'(x_0) = 0,\quad f''(x_0) > 0$

The first derivative is

$f'(x) = 2ax+b \implies f'(x)=0 \iff x = \dfrac{-b}{2a}$

The second derivative is

$f''(x) = 2a$

So, a parabola has a minimum only if $a>0$ (otherwise, the parabola is concave down and it has no lower bound). In that case, the minimum has coordinates

$x = \dfrac{-b}{2a},\quad y = f\left(\dfrac{-b}{2a}\right) = a\left(\dfrac{-b}{2a}\right)^2 + b\left(\dfrac{-b}{2a}\right) + c = \dfrac{4ac-b^2}{4a}$