Discriminant Calculator

The discriminant calculator helps to find the discriminant of the quadratic polynomial as well as higher degree polynomials. You can try this discriminant finder to find out the exact nature of roots and the number of root of the given equation.

What is Discriminant?

In maths, a discriminant is a function of coefficients of the polynomial equation that displays the nature of the roots of a given equation. It is represented by a $Δ$ sign (read as delta). If you have a concern with the term “what does the discriminant tell you”, then keep reading.

Nature of Roots:

In Quadratic equation:

The discriminant of the quadratic equation determines the roots' nature.

If $Δ>0$ and is not a perfect square, then the roots are real and irrational.
If $Δ=0$, then roots are equal and real.
If $Δ<0$, then the roots are imaginary.
If $Δ$ is a perfect square,then the roots are rational.

In Cubic Equation:

The discriminant of the cubic equation determines the roots' nature.

If $Δ>0$, then all the three roots are real.
If $Δ<0$, then one root is real and two roots are complex conjugate roots.
If $Δ=0$, then two roots are equal.

In Quartic Equation:

The discriminant of the quartic equation determines the roots' nature.

If $Δ>0$, then all the four roots are real.
If $Δ<0$, then two roots are real and two roots are complex.conjugate roots.
If $Δ=0$, then two or more roots are equal. There are 6 possibilities:

Three distinct real roots from which one is double.
Two distinct real roots, both are double.
Two distinct real roots, one has a multiplicity of 3.
One real root of a multiplicity of 4.
One real double root and two complex roots.
Pair of double complex roots.

Number of Roots:

In Quadratic Equation:

The discriminant of the quadratic equation determines how many roots are there in an equation.

There are two roots of a quadratic equation.

In Cubic Equation:

The discriminant of the cubic equation determines how many roots are there in an equation.

There are three roots of a cubic equation.

In Quartic Equation:

The discriminant of the quartic equation determines how many roots are there in an equation.

There are four roots of a quartic equation.

The online discriminant calculator shows the nature of roots of the quartic equation and if you want to determine the nature of roots for the cubic and quadratic equations, then this online tool is handy.

Discriminant In Terms of Parabola:

The discriminant of an equation determine the shape of the parabola in a graph,

If $Δ>0$, then the parabola does not cross the x-axis of the coordinate plane.
If $Δ<0$, then parabola intersects the x-axis of the coordinate plane at two points.
If $Δ=0$, then the parabola is tangent to the x-axis of the coordinate plane.

Standard Discriminant Formula:

The standard formula for the following standard polynomial equation is: $$ p(x) = a_nx^n + . . . + a_1x + a_0 $$ the equation has exactly $n$ roots $x_1, . . . , x_n$ (remember that these roots not necessarily all unique! Now, here we figure out the discriminant of $p$ as: $$ D(p) = a_n \text{ }^{2n-2} \prod (x_i - x_j)^2 $$

Where;

the product $\prod$ is taken over all $i < j$

$D(p)$ is referred to as a homogenous polynomial of degree $2 (n-1)$ in the coefficient of $p$
$D(p)$ is said to be as a symmetric function of the roots of $p$, which simply assures that the value of $D(p)$ is independent from the order in which you labeled the roots of $p$

The standard discriminant form for the quadratic, cubic, and quartic equations is as follow,

Quadratic Equation:

The standard discriminant formula for the quadratic equation $ax^2 + bx + c = 0$ is, $$ Δ = b^2-4ac $$

Where,

$a$ is the coefficient of $x^2$.
$b$ is the coefficient of $x$.
$c$ is the constant.

Cubic Equation:

The standard discriminant form for the cubic equation $ax^3 + bx^2 + cx + d = 0$ is,

$Δ=b^2c^2 - 4ac^3-4b^3d-27a^2d^2+18abcd$

Where,

$a$ is the coefficient of $x^3$.
$b$ is the coefficient of $x^2$.
$c$ is the coefficient of $x$.
$d$ is the constant.

Quartic Equation:

The standard discriminant form for the quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0$ is,

$Δ = 256a^3e^3 - 192a^2bde^2 - 128a^2c^2e^2 + 144a^2cd^2e - 27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e$$ - 80abc^2de +18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 +18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2d^2$

Where,

$a$ is the coefficient of $x^4$
$b$ is the coefficient of $x^3$
$c$ is the coefficient of $x^2$
$d$ is the coefficient of $x$.
$e$ is the constant.

Discriminant of Higher Degree Polynomials:

As we know the discriminant of a quadratic equation has only two terms, but as the degree of polynomial increases, the discriminant becomes more complicated.

The discriminant of the cubic equation has 5 terms.
The discriminant of the quartic equation has 16 terms.
The discriminant of the quintic equation have 59 terms.
The discriminant of the sextic equation have 246 terms.
The discriminant of the septic equation has 1103 terms.

How Our Calculator Works:

The discriminant calculator shows you the step-by-step calculations for the given equation problems. It doesn’t matter whether you want to calculate quadratic equation and higher degree polynomials equation, this calculator does all for you!

Inputs:

First of all, you have to select the degree of polynomial from the dropdown of this tool in which you want to find out the discriminant.
Then, enter the coefficient values for the selected equation. (enter the values according to the selected degree of polynomial)
Finally, hit the calculate button

Outputs: The discriminant calculator will find:

The discriminant of the given equation.
Nature of the roots.
Complete calculation of the discriminant.

How to Find The Discriminant Manually (Step-By-Step)?

Let’s have an example of each type of equation and have step by step calculations for each.

For Quadratic Equation:

The formula for the discriminant of quadratic equation is, $$ Δ = b^2-4ac $$

For example:

If we have an equation, $3x^2+2x-9=0$, then find the discriminant?

Solution:

Here,

$a = 3$

$b = 2$

$c = -9$

Putting the values in the given formula,

$Δ = (2)^2-4(3)(-9)$

$Δ = 4+108$

$Δ = 112$

For Cubic Equation:

The formula for the discriminant of cubic equation is, $$ Δ= b^2c^2 - 4ac^3-4b^3d-27a^2d^2+18abcd $$

For example:

Calculate the discriminant of the following equation? $$ 5x^3 + 2x^2 + 8x + 6 = 0 $$

Solution:

Here,

$a = 5$

$b = 2$

$c = 8$

$d = 6$

Putting the values in the given formula,

$Δ=(2)^2(8)^2 - 4(5)(8)^3-4(2)^3(6)-27(5)^2(6)^2+18(5)(2)(8)(6)$

$Δ=(4)(64) - 4(5)(512)-4(8)(6)-27(25)(36)+18(480)$

$Δ=(4)(64) - 4(2560)-4(48)-27(900)+18(480)$

$Δ=256 - 10240-192-24300+8640$

$Δ=8896 - 10240-192-24300$

$Δ=8896 - 10240-24492$

$Δ=8896 - 34732$

$Δ= -25836$

For Quartic Equation:

The formula for the discriminant of quartic equation is,

$Δ=256a^3e^3 - 192a^2bde^2-128a^2c^2e^2 + 144a^2cd^2e -27a^2d^4 + 144ab^2ce^2 - 6ab^2d^2e - 80abc^2de$ $+18abcd^3 + 16ac^4e - 4ac^3d^2 - 27b^4e^2 +18b^3cde - 4b^3d^3 - 4b^2c^3e + b^2c^2²d^2$

For example:

Calculate the discriminant of the following equation?

$2x^4+x^3+3x^2+2x+7=0$

Solution:

Here,

$a = 2$

$b = 1$

$c = 3$

$d = 2$

$e = 7$

Putting the values in the given formula,

$= 256 \times (2)^3 \times (7)^3 - (192) \times (2)^2 \times 1 \times 2 \times (7)^2 - (128) \times (2)^2 \times (3)^2 \times (7)^2 + (144) \times (2)^2 \times 3 \times (2)^2 \times 7 - (27) \times (2)^2 \times (2)^4$

$+(144) \times 2 \times (1)^2 \times 3 \times (7)^2 - (6) \times 2 \times (1)^2 \times (2)^2 \times 7 - (80) \times 2 \times 1 \times (3)^2 \times 2 \times 7 + (18) \times 2 \times 1 \times 3 \times (2)^3$

$+(16) \times 2 \times (3)^4 \times 7 - (4) \times 2 \times (3)^3 \times (2)^2 - (27) \times (1)^4 \times (7)^2+ (18) \times (1)^3 \times 3 \times 2 \times 7 - (4) \times (1)^3 \times (2)^3 $

$-(4) \times (1)^2 \times (3)^3 \times 7 + (1)^2 \times (3)^2 \times (2)^2$

$=(256 \times 8 \times 343) - (192 \times 4 \times 1 \times 2 \times 49) - (128 \times 4 \times 9 \times 49) + (144 \times 4 \times 3 \times 4 \times 7)$

$-(27 \times 4 \times 16) + (144 \times 2 \times 1 \times 3 \times 49) - (6 \times 2 \times 1 \times 4 \times 7 ) - (80 \times 2 \times 1 \times 9 \times 2 \times 7)$

$+(18 \times 2 \times 1 \times 3 \times 8) + (16 \times 2 \times 81 \times 7) -(4 \times 2 \times 27 \times 4 )- (27 \times 1 \times 49)+ (18 \times 1 \times 3 \times 2 \times 7)$

$-(4 \times 1 \times 8) - (4 \times 1 \times 27 \times 7) + (1 \times 9 \times 4)$

$=702464 - 75264 - 225792 + 48384 - 1728 + 42336 - 336 - 20160 + 864 - 18144 - 864 + 1323 + 756 - 32 - 756 + 36$

$= 453087$

References:

The brilliant source provided with: Finding the Discriminant of a Quadratic (Explanation), Repeated Roots and Range of Solutions From the source of studypug: Nature of roots of quadratic equations: