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242 LearnersLast updated on October 21, 2025

When solving quadratic equations to determine the nature of a polynomial equation, we use the concept of the discriminant. This article explains the discriminant, its formula, and its properties.
A quadratic equation is written in the form ax2 + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The roots are the values of x that make the equation true. There are different methods to find the roots of quadratic equations, such as:
In mathematics, the discriminant is a value calculated from the coefficients, used to determine the nature of the roots. It is denoted by Δ or D. The value of the discriminant can be positive, negative, or zero.
The nature of the roots in a quadratic equation is determined by the value of the discriminant. For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant formula is \(D = b^2 - 4ac\).
The discriminant formula for a cubic equation \(ax^3 + bx^2 + cx + d = 0, \quad
D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2\)
As we know, there is a formula to find the value of the discriminant, so by substituting the values in the formula, we find the discriminant. In this section, we will learn how to find the discriminant of a quadratic and cubic equation.
For a quadratic equation \(ax^2 + bx + c = 0\), the value of the discriminant is \(D = b^2 - 4ac\).
We know that for a quadratic equation \(ax^2 + bx + c = 0\), the value of \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
As \(D = b^2 - 4ac\), the value of x can be written as:
\(x = \frac{-b \pm \sqrt{D}}{2a}\)
For example, finding the discriminant of a quadratic equation \(3x^2 - 4x + 8 = 0\)
The formula to find the discriminant is \(D = b^2 - 4ac\)
Here, a = 3, b = -4, and c = 8
\(D = (-4)^2 - 4 \times 3 \times 8\)
= 16 - 96
= -80.
For a cubic equation \(ax^3 + bx^2 + cx + d = 0\) the discriminant can be calculated using the formula:
\(D = 18abcd - 4b^3d + b^2c^2 - 4ac - 27a^2d^2\)
For example, find the discriminant of the cubic equation: \(x^3 - 6x^2 + 11x - 6 = 0\)
To find the discriminant, we use the formula: \(D = b^2 c^2 - 4 a c^3 - 4 b^3 d - 27 a^2 d^2 + 18 a b c d\)
Here, a = 1
b = -6
c = 11
d = -6
\(D = (-6)^2 (11)^2 - 4(1)(11)^3 - 4(-6)^3(-6) - 27(1)^2(-6)^2 + 18(1)(-6)(11)(-6)\)
\(D = 4356 -5324 = -968; -968 - 5184\)
\(D = -6152; -6152 - 972 = -7124 \)
\(D = -7124 + 7128\)
D = 4
The roots of a quadratic equation are calculated using the formula:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
The quadratic equation can be written as:
\(x = \frac{-b \pm \sqrt{D}}{2a}, \quad \text{where } D = b^2 - 4ac\)
The discriminant is used for the nature of the roots, which means what type of numbers the root can be. A root can be real, rational, irrational, or imaginary.
The value of the discriminant can predict the nature of the roots. The value of D can be positive, negative, or zero. Here are the differences between positive, zero, and negative discriminants.
|
Positive Discriminant (D > 0) |
Negative Discriminant (D < 0) |
Zero Discriminant (D = 0) |
|
When D > 0, x = -b ± √D/2a |
When D < 0, x = -b ± √D/2a |
When D = 0, x = -b ± √0/2a |
| A positive discriminant has two distinct roots | A negative discriminant has two complex conjugate roots | A zero discriminant results in one repeated real root. |
|
When D > 0, the roots are two distinct real root |
When D < 0, the root is a complex number |
When D = 0, the root is a real number, and it is repeated |
The discriminant is a powerful tool in quadratic equations that helps predict the nature, type, and sometimes even the value of roots without fully solving the equation. These tips and tricks make it quick and easy to use.
Students make errors when solving the quadratic equation and finding the discriminant. Mostly, students often repeat the same errors. In this section, let’s learn some common mistakes and the ways to avoid them.
In mathematics, the discriminant is a fundamental concept used mainly in quadratic equations. It is also used in the fields of engineering, physics, computer graphics, economics, etc. Here are the applications of discriminant in our real life.
Find the discriminant of x square + 4x + 4 = 0
The value of the discriminant is 0
The value of the discriminant can be calculated using the formula:
D = b2 - 4ac
Here, a = 1, b = 4, c = 4
So, D = 42 - 4 (1)(4)
= 16 - 16 = 0
A garden fence is planned based on the equation x square - 2x - 3 = 0. Find the discriminant, and what is the nature of the solutions.
The discriminant is 16, and the equation has two real solutions
The formula for the discriminant: D = b2 - 4ac
Here, a = 1, b = -2, c = -3
So, D = (-2)2 - 4 × 1 × -3
= 4 - (-12)
= 4 + 12
= 16
As D > 0, it has two distinct real solutions.
Find the discriminant of x^2 +2x + 5 = 0
The value of D is -16
The formula of the discriminant is: D = b2 - 4ac
Here, a = 1, b = 2, c = 5
Then D = 22 - 4(1)(5)
= 4 - 20
= -16
Using the discriminant to determine the nature of its roots of the quadratic equation: x square + x + 1 = 0
Here, the value of D is -3, as the value of D is negative, it has two complex roots
To find the nature of the roots, we use the discriminant.
The discriminant formula is: D = b2 - 4ac
D = 12 - 4 × 1 × 1
= 1 - 4
= -3
When the value of D is negative, the solution has two complex roots.
Find the value of discriminant of x^2 + 5x + 6 = 0
The value of the discriminant is 1
The discriminant formula is: D = b2 - 4ac
Here, a = 1, b = 5, and c = 6
D = 52 - 4 (1)(6)
= 25 - 24
= 1
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.






