# Discriminant

 1 Discriminant in Math 2 Discriminant Formula 3 Relation between Discriminant & Nature of Roots 4 Discriminant Calculator (with Graph) 5 Important Notes on Discriminant 6 Solved Examples on Discriminant 7 Practice Questions on Discriminant 8 Challenging Questions on Discriminant 9 Maths Olympiad Sample Papers 10 Frequently Asked Questions (FAQs)

## 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 $$a,b,c$$ $$\text{ and }$$ $$d$$.
$\Delta\!\!=\!b^{2} c^{2}\!\!-\!4 a c^{3}\!\!-\!4 b^{3} d\!\!-\!27 a^{2} d^{2}\!\!+\!\!18 a b c d$

## 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.

## 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
1. The discriminant of a quadratic equation $$\mathbf{ax^2+bx+c=0}$$ is $$D \text{ OR } \Delta = b^2-4ac$$
(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.

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.

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.

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.

## 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.

(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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