Discriminant

Discriminant in Math

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.

Discriminant symbol: discriminant is represented by delta or D


Discriminant Formula

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 \(\mathbf{\Delta=-15}\)

Relation between Discriminant and Nature of the Roots

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.

Relation between discriminant and the nature of the roots is shown using graphs.


Discriminant Calculator (with Graph)

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.

 
important notes to remember
Important Notes
  1. The discriminant of a quadratic equation \(\mathbf{ax^2+bx+c=0}\) is \(D \text{ OR } \Delta = b^2-4ac\)
  2. A quadratic equation has:
    (i) two unequal real roots when \(D>0\)
    (ii) only one real root when \(D=0\)
    (iii) no real roots or two complex roots when \(D<0\)

Solved Examples

Here are some discriminant examples and their solutions.

Example 1

 

 

Find the discriminant of \(\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0\)

Solution:

Comparing the given equation with \(ax^2+bx+c=0\),

\[ \begin{aligned} a&= \sqrt{3}\\[0.2cm]b &=10\\[0.2cm]c&=-8\sqrt{3} \end{aligned} \]

The discriminant of the given equation is,

\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (10)^2 -4( \sqrt{3})(-8\sqrt{3})\\[0.2cm] &=100+96\\[0.2cm]&=196 \end{aligned}\]

\(\mathbf{D}\) or \(\mathbf{\Delta=196}\)
Example 2

 

 

Determine whether the following equation has two real roots, one real root or no real roots.

\[3x^2 - 5x -7=0\]

Solution:

Comparing the given equation with \(ax^2+bx+c=0\),

\[ \begin{aligned} a&=3\\[0.2cm]b &=-5\\[0.2cm]c&=-7\end{aligned} \]

The discriminant of the given equation is,

\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (-5)^2 -4( 3)(-7)\\[0.2cm] &=25+84\\[0.2cm]&= \sqrt{109}\end{aligned}\]

Since the discriminant is positive here,

The number of real roots \(= 2\)

Let us check it using the graph.

A quadratic equation with two x-intercepts is shown on a graph.

It has two x-intercepts which means that the number of real roots is \(2\).

Example 3

 

 

Find the discriminant of the following equation:

\[9 z^{2}-6 b^{2} z-\left(a^{4}-b^{4}\right)=0\]

Solution:

Comparing the given equation with \(Ax^2+Bx+C=0\),

\[ \begin{aligned} A&=9\\[0.2cm]B &=-6b^2\\[0.2cm]C&=-(a^4-b^{4})\end{aligned} \]

The discriminant of the given equation is,

\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (-6b^2)^2 -4( 9)(-(a^4-b^{4}))\\[0.2cm] &=36b^4 +36a^4-36b^4\\[0.2cm]&= 36a^4\end{aligned}\]

\(\mathbf{D}\) or \(\mathbf{\Delta=36a^4}\)

Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
Challenge your math skills
Challenging Questions
 
1. Identify the discriminant of \(\mathbf{x^{2}+5 x-\left(p^{2}+p-6\right)=0}\) from the following:
  \(\begin{aligned}\mathbf{a.D\!=\!(2 p\!+\!1)}\end{aligned}\) \(\begin{aligned}\mathbf{b.D\!=\!(2 p\!+\!1)^{2}}\end{aligned}\)
  \(\begin{aligned}\mathbf{c.D\!=\!4 p^{2}}\end{aligned}\) \(\begin{aligned}\mathbf{d.D\!=\!(2 p\!-\!7)^{2}}\end{aligned}\)
     
2. Find the positive value of \(k\) such that the equation \(\mathbf{k x^{2}-24 x+2=0}\) has only one real solution.

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is the discriminant formula?

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

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

2. What is a discriminant example?

Example

Find the discriminant of \(\sqrt{3} x^{2}+11 x+10 \sqrt{3}=0\)

Solution

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

\[a=\sqrt{3}\\[0.2cm]b=11\\[0.2cm]c=10\sqrt{3}\]

The discriminant of the given equation is:

\[ \begin{aligned} D \text{ OR }\Delta &= b^2-4ac\\[0.2cm] &= (11)^2 -4(\sqrt{3})(10\sqrt{3})\\[0.2cm] &=121-120\\[0.2cm] &=1 \end{aligned}\]

Thus, the discriminant of the given equation is:

\(\mathbf{D}\) or \(\mathbf{\Delta=1}\)

3. How do you use the discriminant formula?

We can use the discriminant to find the nature of the roots.

A quadratic equation has:
(i) two unequal real roots when \(D>0\)
(ii) only one real root when \(D=0\)
(iii) no real roots or two complex roots when \(D<0\)

  
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