Discriminant of a Quadratic Equation

Quadratic was derived from Quadratus, a Latin word that describes the second degree or the operation of squaring.

We define the Quadratic Equation in the following generalized format.

f(x)=ax²+bx+c with ‘a’ not equals to zero. Here, ax² is a Quadratic term, bx is linear term (bx¹) and c is the constant term (cx⁰).

Here, ‘a’ is not equal to zero. Because, assume if ‘a’ is equal to zero, that is; the coefficient of the highest degree is zero, then f(x) won’t be a Quadratic Equation.

When ‘a’ is not equal to 0, then there we have two possibilities for the value of ‘a’.

1. a >0
2. a <0

When a >0, the function will have a minimum value, that is, for a particular value of x, the function will give a minimum value.

The coefficient of the highest degree member/Quadratic term is greater than zero

When a <0, the function will have a maximum value, that is, for a particular value of x, the function will give a maximum value.

The coefficient of the highest degree member/Quadratic term is lesser than zero

For a given Quadratic Equation (f(x)=ax²+bx+c), the Discriminant shall be given as,

This is derived from the root of x (the independent variable) in a Quadratic Equation when it is equalized to zero.

f(x)=ax²+bx+c=0

Here, we all know that, if negative value is placed inside a square root, that will give an imaginary value. That is; it is not possible to square a value (multiply it times itself) and arrive at a negative value. To address this problem, we have created a new number, i, which refers an “imaginary number”, and this is not in the set of “Real Numbers”. This new number was viewed with much skepticism. Also i²=(-1).

Let us alter the solution for x as follows,

So now, we have 3 possibilities for the value of Δ for each range of ‘a’.

1. Δ>0
2. Δ=0
3. Δ<0

When a >0 and Δ>0,

x can has

or

Both are real values and this shows the function will has two real roots and the function will cut the x axis in two distinct points.

When a >0 and Δ=0,

x can has

as the only real solution. This shows the function will has only one real root (coinciding two real roots) and will touch the x axis at a distinct place.

When a >0 and Δ<0,

x can has

or

Both are not real values (that is imaginary values) and this shows the function will never has real roots and the function will never cut the x axis. Simply, the function will has two imaginary roots.

Check the following graphs for the graphical understanding.

I hope you are now be able to figure out how it works when a <0.