A quadratic equation is an equation of the form

$$ax^2 + bx + c = 0 \tag{1} $$

where

$ a,b,c \in \mathbb{R}$ and $a \ne 0 $.

Let us assume that we are trying to solve this equation for real numbers only.

The value

$$D = b^2 - 4ac \tag{2} $$

is called discriminant of the equation $(1)$.

Case #1:

When $D \gt 0$, there are two distinct real solutions to $(1)$ and they are:

$$x_{1,2} = {-b \pm \sqrt{b^2-4ac} \over 2a} \tag{3} $$

Case #2:

When $D = 0$, there is one real solution to $(1)$ which is:

$$x = {-b \over 2a} \tag{4} $$

Case #3:

When $D \lt 0$, there are no real solutions to $(1)$.

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