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Last updated on October 21, 2025

Discriminant

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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.

Discriminant for US Students
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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.
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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.  

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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, \(ax^2 + bx + c = 0\), the value of the discriminant depends on the coefficients a, b, and c, and it is used to understand the type of solutions. 
     
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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 \(ax^2 + bx + c = 0\), the discriminant formula is \(D = b^2 - 4ac\)


The discriminant formula for a cubic equation \(ax^3 + bx^2 + cx + d = 0, \quad D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2\)

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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. 

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Discriminant of a Quadratic 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. 
 

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Discriminant of a Cubic Equation

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 

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What is the Nature of Roots of Discriminant?

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. 

 

 

  •  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. 
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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

 

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Tips and Tricks of Discriminant

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.

 

  1. Discriminant is D = b2 −4ac; identify a, b, c carefully to avoid mistakes.

     
  2. If D > 0 is two real roots, D = 0 is one real roots, D < 0 is two complex roots.

     
  3. If D is a perfect square, roots are rational; otherwise, they are irrational.

     
  4. A negative D means no real roots, saving you from solving the quadratic.

     
  5. Perfect-square D suggests easy factorization; non-perfect-square D needs the quadratic formula.
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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

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Using the wrong discriminant formula
 

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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

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Using the wrong coefficient values
 

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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

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Thinking all roots are real
 

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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

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Considering the discriminant as always positive
 

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Some students mistakenly assume that the discriminant is always positive. So, make sure you are not assuming; instead, always verify the answer.

Mistake 5

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Ignoring the graphical meaning

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Some students focusing only on numbers, ignoring the parabola’s behavior. Sketch a simple graph to visualize how Δ affects intersections with the x-axis.

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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. 

 

 

  1. Engineering and Physics: The discriminant helps determine if a quadratic equation describing motion or forces has real solutions, predicting outcomes like projectile landing points.

     
  2. Finance and Economics: Used to analyze profit and cost functions modeled by quadratics, the discriminant shows whether break-even points exist and how many there are.

     
  3. Architecture and Design: In structural design, parabolic curves often model arches or bridges; the discriminant helps check if certain dimensions produce real feasible intersections.

     
  4. Computer Graphics: When calculating ray-object intersections in 3D graphics, the discriminant indicates whether rays intersect surfaces, touch tangentially, or miss entirely.

     
  5. Robotics and Navigation: Quadratic equations often arise in trajectory planning. The discriminant helps decide whether a robot can reach a specific point or if the path is impossible.
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Solved Examples of Discriminant

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Problem 1

Find the discriminant of x square + 4x + 4 = 0

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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
 

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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.

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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.
 

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Problem 3

Find the discriminant of x^2 +2x + 5 = 0

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The value of D is -16
 

Explanation

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
 

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Problem 4

Using the discriminant to determine the nature of its roots of the quadratic equation: x square + x + 1 = 0

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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. 
 

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Problem 5

Find the value of discriminant of x^2 + 5x + 6 = 0

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The value of the discriminant is 1
 

Explanation

The discriminant formula is: D = b2 - 4ac


Here, a = 1, b = 5, and c = 6


D = 52 - 4 (1)(6) 


= 25 - 24 


= 1
 

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FAQs on Discriminant

1.What is discriminant?

The discriminant of a polynomial is the value derived from the coefficients, it is used to determine the nature of the roots 

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2.What is the formula for the discriminant?

The discriminant formula is different for a quadratic and cubic equation. The discriminant of a quadratic equation ax2 + bx + c is D = b2 - 4ac. For a cubic equation ax3 + bx2 + cx + d = 0, the discriminant is: D = b2c2 - 4ac3 - 4b3d - 27a2d2 + 18abcd  

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3.What does the discriminant tell?

The value of the discriminant is used to determine the nature of the equation. 
If D > 0, the root has two real roots
If D < 0, the root has two complex roots
If D = 0, the equation has one real root, and it is repeated
 

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4.What is the nature of the roots of a quadratic equation?

The discriminant of the quadratic equation determines the nature of its roots, whether they are real or imaginary. 
 

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5.Can the discriminant be a decimal or a fraction?

Yes, the discriminant can be a decimal or a fraction. 

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6.How can a parent explain a negative discriminant to their child?

Parents can tell their child that a negative discriminant means the roots are complex, valid numbers that cannot be seen on a regular number line.

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7.How can a parent help their child remember the discriminant formula?

A parent can guide their child to memorize D = b2−4ac with simple phrases like “b squared minus four a c” and practice small examples together.

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8.How can a parent show their child that D < 0 doesn’t mean failure?

Parents can explain that complex roots are just another type of solution, valid in math, helping the child see that a negative discriminant is not “wrong.”

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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.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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