Discriminant

Table of Contents 

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

Discriminant Symbol

The symbol of discriminant is either \(\mathbf{D}\) or \(\mathbf{\Delta}\)

For example,

• the discriminant of a quadratic equation \(ax^2+bx+c=0\) is in terms of \(a,b,\) \( \text{ and }\) \(c\).
  \[\Delta \text{ OR } D=b^2-4ac\]
• the discriminant of a cubic equation \(ax^3+bx^2+cx+d=0\) is in terms of \(a,b,c\) \( \text{ and }\) \(d\).
  \[\Delta\!\!=\!b^{2} c^{2}\!\!-\!4 a c^{3}\!\!-\!4 b^{3} d\!\!-\!27 a^{2} d^{2}\!\!+\!\!18 a b c d\]

We usually study about the discriminant of a quadratic equation.

The discriminant,  \(\mathbf{D}\) or \(\mathbf{\Delta}\), of a quadratic equation \(ax^2+bx+c=0\) is:

\(\mathbf{D}\) or \(\mathbf{\Delta=b^{2}-4 a c}\)

Do you recall using \(b^2-4ac\) earlier?

Yes, it is a part of the quadratic formula:

\[x=\dfrac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\]

Here, the expression that is inside the square root is called the discriminant of the quadratic equation.

The quadratic formula in terms of the discriminant is:

\[x=\dfrac{-b \pm \sqrt{D}}{2 a}\]

The discriminant can be either positive or negative or zero.

Discriminant Examples

Here is a discriminant example.

Example

Find the discriminant of \(2x^2+3x+3=0\)

Solution

Compare the given expression with \(ax^2+bx+c=0\),

\[a=2\\b=3\\c=3\]

The discriminant of the given equation is:

\[ \begin{aligned} D \text{OR}\Delta &= b^2-4ac\\[0.2cm] &= 3^2 -4(2)(3)\\[0.2cm] &=9-24\\[0.2cm] &=-15 \end{aligned}\]

Thus, the discriminant of the given equation is:

\(\mathbf{D}\) or \(\mathbf{\Delta=-15}\)

The roots of a quadratic equation \(ax^2+bx+c=0\) are the values of \(x\) that satisfy the equation.

They can be found using the quadratic formula:

\(x=\dfrac{-b \pm \sqrt{D}}{2 a}\)

Though we cannot find the roots by just using the discriminant, we can determine the nature of the roots as follows:

• If \(D>0\), the quadratic equation has two different real roots:
  \[ \dfrac{-b \pm \sqrt{ \text{Positive number}}}{2 a}\]
  gives two roots
   
• If \(D=0\), the quadratic equation has only one real root:
  \[ \dfrac{-b \pm \sqrt{0}}{2 a}=  \dfrac{-b }{2 a}\]
  is the only root
   
• If \(D<0\), the quadratic equation has no real roots. i.e., it has two complex roots:
  \[ \dfrac{-b \pm \sqrt{ \text{Negative number}}}{2 a}\]
  gives two complex roots.

This is because the square root of a negative number leads to an imaginary number. i.e., \(\sqrt{-1} = i\)

A root is nothing but the x-coordinate of the x-intercept.

The graph of a quadratic equation in each of these 3 cases can be as follows.

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Here is the "Discriminant Calculator".

You can move the sliders to change the coefficients of a quadratic expression.

Then it shows the corresponding discriminant.and the graph.

It also shows the number of roots depending on the discriminant.