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220 LearnersLast updated on September 26, 2025
Discriminant
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.
What is a Quadratic Equation?
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:
- Factoring
- Completing the square
- The quadratic formula.
What is Discriminant in Math?
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.
What are the Properties of the Discriminant?
- When finding the roots of a quadratic equation, the discriminant plays a major role.
- The nature of the roots are real and distinct, real and equal or complex, is determined by the value of the discriminant.
- For a quadratic equation, ax2 + bx + c = 0, the value of the discriminant depends on the coefficients a, b, and c, and it is used to understand the type of solutions.
What is the Formula for the Discriminant?
The nature of the roots in a quadratic equation is determined by the value of the discriminant. For a quadratic equation ax2 + bx + c = 0, the discriminant formula is D = b2 - 4ac.
The discriminant formula for a cubic equation ax3 + bx2+ cx + d = 0 is D = D = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d²
How to Find Discriminants?
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.
Discriminant of a Quadratic Equation
For a quadratic equation ax2 + bx + c = 0, the value of the discriminant is D = b2 - 4ac.
We know that for a quadratic equation ax2 + bx + c = 0, the value of x = -b ± √b2 - 4ac/2a.
As D = b2 - 4ac, the value of x can be written as:
x = -b ± √D/2a
For example, finding the discriminant of a quadratic equation 3x2 - 4x + 8 = 0
The formula to find the discriminant is D = b2 - 4ac
Here, a = 3, b = -4, and c = 8
D = (-4)2 - 4 × 3 × 8
= 16 - 96
= -80.
Discriminant of a Cubic Equation
For a cubic equation ax3 + bx2 + cx + d = 0 the discriminant can be calculated using the formula:
D = 18abcd - 4b3d + b2c2 - 4ac - 27a2d2
For example, find the discriminant of the cubic equation: x3 - 6x2 + 11x - 6 = 0
To find the discriminant, we use the formula: D = b2c2 - 4ac3 - 4b3d - 27a2d2 + 18abcd
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
What is the Nature of Roots of Discriminant?
The roots of a quadratic equation are calculated using the formula:
x = -b ± √b2 - 4ac/2a
The quadratic equation can be written as:
x = -b ± √D/2a, as D = b2 - 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.
- If the Discriminant is Positive: If the discriminant is positive, D > 0, then there are two real roots for the quadratic equation. In other words, if D > 0, x becomes x = (-b ± √positive number) / 2a, as the square root of a positive number is always a real number.
- If the Discriminant is Negative: When the discriminant of a quadratic equation is negative (D < 0), then it has two complex conjugate roots. If D < 0, the value of x = (-b ± √negative number) / 2a. The square root of a negative number always results in an imaginary number.
- If the Discriminant is Equal to Zero: If the discriminant of a quadratic equation is zero, that is D = 0, then the root of the equation is a real number. Because when D = 0, x = (-b ± √0) / 2a and the square root of 0 is 0, so x = -b/2a.
What is the Difference Between Positive, Zero, and Negative Discriminants?
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 |
Real-world Applications of Discriminant
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.
- In construction, it helps assess the stability of structures by predicting the potential issues and optimizing the design for safety.
- To understand the drug concentrations, pharmacologists use quadratic equations. The value of the discriminant is used to analyze dosage optimization.
- In sports like javelin throw and disc throw, to predict the projectile motion, we can use the discriminant.
Common Mistakes and How to Avoid Them in Discriminant
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.
Mistake 1
Using the wrong discriminant formula
Using the wrong discriminant formula, that is, D = b2 + 4ac instead of D = b2 - 4ac. So, memorize the quadratic equation: x = (-b ± √(b² - 4ac)) / 2a and D = b2 - 4ac.
Mistake 2
Using the wrong coefficient values
When finding the value of the discriminant, students sometimes misidentify the values of a, b, and c. So first write down the value of a, b, and c, and then substitute the values in the equation.
Mistake 3
Thinking all roots are real
Students often assume that the roots of a quadratic equation are always real, which is incorrect. But it is not true, as when D < 0, the roots are complex numbers. So always remember that the roots can be both real and imaginary.
Mistake 4
Considering the discriminant as always positive
Some students mistakenly assume that the discriminant is always positive. So, make sure you are not assuming; instead, always verify the answer.
Mistake 5
Thinking the discriminant is the solution
Some students incorrectly believe that the discriminant itself is the solution of the equation. The discriminant is used only to determine the nature of the roots; to find the roots of the equation, use factoring or the quadratic formula.
Solved Examples of Discriminant
Problem 1
Find the discriminant of x square + 4x + 4 = 0
The value of the discriminant is 0
Explanation
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
Problem 2
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
Explanation
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.
Problem 3
Find the discriminant of 6x square + x + 1 =0
The value of D is -23
Explanation
The formula of the discriminant is: D = b2 - 4ac
Here, a = 6
b = 1
c = 1
Then D = 12 - 4 × 6 × 1
= 1 - 24
= -23
Problem 4
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
Explanation
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.
Problem 5
Find the value of discriminant of 3x square + 6x + 3 = 0
The value of the discriminant is 0
Explanation
The discriminant formula is: D = b2 - 4ac
Here, a = 3, b = 6, and c = 3
D = 62 - 4 × 3 × 3
= 36 - 36
= 0
FAQs on Discriminant
1.What is discriminant?
2.What is the formula for the discriminant?
3.What does the discriminant tell?
4.What is the nature of the roots of a quadratic equation?
5.Can the discriminant be a decimal or a fraction?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How do technology and digital tools in United States support learning Algebra and Discriminant ?
8.How can cultural or local activities in United States support learning Algebra topics such as Discriminant ?
9.Does learning Algebra support future career opportunities for students in United States?
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Jaskaran Singh Saluja
About the Author
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
