Discriminants and determining the no. of real roots of a quadratic equation.

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What is discriminant in math?

Discriminant in math is a function of the coefficients of the polynomial. Discriminant Symbol  

The symbol of discriminant is   “D“. 

To find the number of real roots of quadratic equation first we should know what is the quadratic equation?   

A quadratic equation is that in which the maximum power of variable is 2  

General Form of Quadratic Equation

A quadratic equation is in the form: \displaystyle ~a{{x}^{2}}~+~bx~+~c~=~0~

The discriminant of a quadratic equation \displaystyle ~a{{x}^{2}}~+~bx~+~c~ is in terms of a , b , and c is  

Let us clear the concept of discriminant with help of some examples: 

Example1 

Find the discriminant of  \displaystyle ~2{{x}^{2}}~+~3x~+~3~=~0~

Sol:  Compare the given expression with \displaystyle ~a{{x}^{2}}~+~bx~+~c~=~0~

a=2

b=3

c=3

The discriminant of the given equation is 

By applying the formula \displaystyle D~=~{{b}^{2}}-~~4ac

                      =  \displaystyle ~{{3}^{2}}~-~4(2)(3)~

                      = 9 – 24

                      = − 15 

Thus, the discriminant of the given equation is  − 15

There are three cases for the discriminant: 

Case 1: 

\displaystyle ~{{b}^{2}}~-~4ac~>~0~

If the discriminant is greater than zero, this means that the quadratic equation has two real, distinct roots. 

Let’s understand that with an example: 

\displaystyle ~{{x}^{2}}~-~5x~+~2~=~0~

Here, a = 1   , b=  – 5 ,  c= 2 

Discriminant , \displaystyle D~=~{{b}^{2}}-~~4ac

                     =  \displaystyle ~{{(-5)}^{2}}~-~4(1)(2)~

                      = 17 

Discriminant  is greater than zero i.e. 17 > 0. 

Therefore, the roots of the above equation will be real and distinct.  

Case 2: 

When, \displaystyle ~{{b}^{2}}~-~4ac~<~0~

If the discriminant is less than zero then the quadratic equation has no real roots.  

Example: 

\displaystyle ~3{{x}^{2}}~-~2x~+~1~=~0~

Here,  a= 3  ,   b = 2  , c= 1  

Discriminant , \displaystyle D~=~{{b}^{2}}-~~4ac

                              = \displaystyle ~{{(2)}^{2}}~-~4(3)(1)~

                              = 4  −  12 

                               =  − 8 which is < 0 

In above equation value of the discriminant is less than zero. Therefore, the roots of this quadratic equation are not real.  

Case 3: 

 When, D or \displaystyle ~{{b}^{2}}~-~4ac~=~0~

If the discriminant   is equal to zero, this means that the quadratic equation has two   real, identical roots .  

Example : 

\displaystyle ~{{x}^{2}}~+~2x~+~1~=~0~

Comparing the above equation with general quadratic equation  \displaystyle ~a{{x}^{2}}~+~bx~+~c~=~0~

We get a = 1, b  =  2,  c  =  1

Discriminant , \displaystyle D~=~{{b}^{2}}-~~4ac

                               = \displaystyle ~{{(2)}^{2}}~-~4(1)(1)~

                               = 4 − 4

                               = 0

In the above equation discriminant is zero therefore the above equation has two real, identical roots.  

So, from above discussion we can conclude that  

• • If discriminant <  0,  there are  2  imaginary roots . Imaginary roots are those which are in the form of i which is known as ita in mathematics as discriminant is negative.  

• • If discriminant > 0, there are 2 real, distinct roots.  
  • If discriminant = 0, there is one real root and the value of that root is   \frac{{-b}}{{2a}}

So, the discriminant of a quadratic polynomial provides information about the properties of the roots of the polynomial. 

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