In the case of quadratic equations, the discriminant is commonly employed to determine the nature of the roots. Though determining a discriminant for any polynomial is difficult, we may use formulas to get the discriminant of quadratic and cubic equations.

This online calculator calculates the discriminant of the quadratic polynomial, as well as higher degree polynomials

In algebra, a polynomial’s discriminant is a polynomial function of its coefficients that allows some features of the roots to be deduced without computing them.

You’re presumably familiar with the discriminant formula for the quadratic polynomial ax²+bx+c, which is b²-4ac, and you use it to compute the roots.

The discriminant, on the other hand, allows us to deduce some root attributes without having to compute them. It is zero in the case of a quadratic polynomial if and only if it has a double root. If the polynomial has two real roots, it is positive; if the roots are complex, it is negative.

The discriminant is computed using the calculator below, and you can read more about discriminants directly beneath it.

Learn more about the discriminant, including its formulas, as well as the relationship between the discriminant and the nature of the roots.

Table of Contents

**What is Discriminant Calculator?**

The discriminant calculator is a free online application that calculates the discriminant value for supplied quadratic equation coefficients. Our online discriminant calculator tool speeds up and simplifies computations by displaying the result in a fraction of a second. Thank you for visiting our discriminant calculator. When you need to quickly find the discriminant of polynomials with two to five degrees, such as quadratic, cubic, quartic, or quintic polynomials, use it shamelessly. Is it necessary to learn about the discriminant in math? Or how does the discriminant’s math formula look? Please scroll down!

Read Also: Double Integral Calculator: Definition, Examples

The discriminant is defined mathematically and how to find the discriminant of a polynomial is explained in the sections that follow.

We devote an entire section on the discriminant of a quadratic equation, because we know how frequently they are employed. However, we don’t stop at the second degree, though. We go all the way up to the fifth.

**Discriminant Calculator Step By Step**

The following is how to use the discriminant calculator:

Step 1: Fill in the input fields with coefficient values such as “a,” “b,” and “c.”

Then, step 2: To retrieve the result, click the “Solve” button.

Then, step 3: In the output field, the discriminant value will be presented.

**Discriminant Calculator Definition**

In arithmetic, a polynomial’s discriminant is a function of the polynomial’s coefficients. It’s useful for figuring out what kind of solutions a polynomial equation has without having to locate them. It discriminates between the equation’s solutions (as equal and unequal; real and nonreal), therefore the name “discriminant.” It’s commonly represented by ∆ or D. The discriminant’s value can be any real number (i.e., either positive, negative, or 0).

A polynomial of degree n has the following discriminant:

A(x)=a_n x^n +a_{n-1} x^{n-1} + … +a_1 x + a_0 can be defined either in terms of the quotient of the resultant or in terms of the roots.

In terms of the roots, the discriminant is equal to

Disc_x (A) = a(_n)^{2n-2} Π_i g.t. j (r_i – r_j)^2 =(-1)^{½n(n-1)} {a_n}^{2n-2} Π_{i ≠ j} ( r_i – r_j )

Without knowing anything about the discriminant, one can derive the formula for the quadratic equation. The b²-4ac is then obtained by plugging derived formulas for the roots into the specification above.

The discriminant is equivalent to in terms of the resultant’s quotient.

Disc_x (A)= [(-1)^{n(n-1)/2}] / A_n × Res_x (A, A’)

where Res is the first derivation of A’ and the resultant of A. The determinant of the Sylvester matrix of A and A’ is the outcome.

The A is ax²+bx+c in the case of a quadratic polynomial, and the A’ is 2ax+b. You’ll get the b²-4ac if you write out the Sylvester matrix for these two polynomials and derive the determinant.

**Discriminant Calculator with steps**

A discriminant is a function of polynomial equation coefficients that shows the nature of the roots of a given equation. A ∆ symbol is used to represent it (read as ∆). Continue reading if you are concerned with the phrase “what does the discriminant tell you.”

Also, use this online quadratic formula calculator to solve the quadratic problem using the quadratic formula or the complete the square approach.

**Discriminant Calculator** **Nature of Roots:**

### In Quadratic equation:

The nature of the roots is determined by the discriminant of the quadratic equation.

- The roots are real and irrational if ∆ is greater than 0 and is not a perfect square.
- Then, the roots are equivalent and real if ∆ = 0.
- Then, the roots are imaginary if ∆ is less than zero.
- Similarly, the roots are reasonable if ∆ is a perfect square.

### In Cubic Equation:

The nature of the roots is determined by the discriminant of the cubic equation.

- All three roots are real if ∆ is greater than zero.
- If ∆ is less than 0, one root is real, while the other two are complex conjugate roots.
- The two roots are equal if ∆ = 0.

### In Quartic Equation:

The nature of the roots is determined by the discriminant of the quartic equation.

- If ∆ is greater than 0, the four roots are all real.
- Two roots are real and two roots are complicated if ∆ is less than 0.
- roots conjugate
- If ∆ = 0, two or more roots must be equal. There are six options available:
- There are three unique real roots, one of which is double.
- Both true roots are twofold and separate.
- There are two separate real roots, one of which has a multiplicity of three.
- One of the four real roots of a multiplicity of four.
- Two complex roots and one actual double root
- A pair of roots with double complexes.

**Discriminant Calculator** **Number of Roots:**

### In Quadratic Equation:

The number of roots in a quadratic equation is determined by the discriminant of the equation.

- The roots of a quadratic equation are two.

### In Cubic Equation:

The cubic equation’s discriminant determines how many roots are there in an equation.

- The roots of a cubic equation are three.

### In Quartic Equation:

The quartic equation’s discriminant determines how many roots are there in an equation.

- The roots of a quartic equation are four.

The online discriminant calculator displays the nature of roots in quartic equations, and it can also be used to determine the nature of roots in cubic and quadratic equations.

**Discriminant Calculator In Terms of Parabola**

The shape of the parabola in a graph is determined by the discriminant of an equation.

- The parabola does not intersect the x-axis of the coordinate plane if ∆ is bigger than 0.
- If ∆ is less than zero, the parabola intersects the coordinate plane’s x-axis twice.
- If ∆ = 0, the parabola is tangent to the coordinate plane’s x-axis.

**Properties of Discriminant Calculator**

We may deduce some fundamental aspects of discriminants from the arithmetic formula for discriminants presented in the preceding sections.

Assume D(p) is the discriminant of p, as defined above. D(p) values are as follows:

Firstly, D(p) is always a real integer since p is a real polynomial.

Then, D(p) = 0 if and only if p has a multiple root.

Similarly, D(p) is bigger than 0 if and only if p’s number of non-real roots is a multiple of four (zero included).

In particular, D(p) is bigger than 0 if all the roots are real and simple.

In terms of discriminant invariance, we have:

Under translation, D(p) is invariant:

If q(x) = p(x + a), then D(q) = D(p).

D(p) is invariant (up to scaling) under homothety:

If q(x) = p(a * x), then D(q) = aⁿ⁽ⁿ⁻¹⁾D(p).

**Discriminant Calculator of a quadratic equation formula**

Consider the quadratic polynomial ax² + bx + c. The formula for its discriminant is:

b² – 4ac

The square root of this discriminant appears in the formula for quadratic polynomial roots, as we all recall:

[-b ± √(b² – 4ac)] / (2a)

We can derive the following from the sign of the discriminant without computing the roots:

D > 0 if the polynomial has two different real roots; D 0 if the polynomial has a pair of conjugate complex roots; and D = 0 if the polynomial has a double root.

Furthermore, if the coefficients a, b, and c are rational, both polynomial roots are rational if and only if D is the square of a rational number.

D is greater than 0 in terms of the parabola in the real plane if, and only if, the parabola does not intersect the horizontal axis;

If the parabola contacts the horizontal axis at two points, D is less than 0; if the parabola touches (is tangent to) the horizontal axis, D is equal to 0.

Remember that quadratic equations can be solved by completing the square!

**Discriminant Calculator of higher degree polynomials**

A general quadratic discriminant has only two terms, as we’ve shown. The discriminant, on the other hand, grows more difficult as the degree of the polynomial increases:

- A general cubic discriminant has five terms.
- The discriminant of a quartic, on the other hand, has 16 terms.
- Then, the discriminant of a quintic, on the other hand, has 59 words.
- Then, a sextant’s discriminant has 246 words.
- Similarly, a sceptic’s discriminant, on the other hand, has 1103 words.

The OEIS sequence A007878 is formed by these numbers. You can find a few more terms there.

**Discriminant Calculator of a cubic polynomial**

Consider the cubic polynomial ax³ + bx² + cx + d. Its discriminant formula reads

b²c² – 4ac³ – 4b³d – 27a²d² + 18abcd.

We have,

D is larger than 0 if there are three different real numbers in the roots.

Similarly, D is less than 0 if there is one real root and two complex conjugate roots.

Then, D = 0 if at least two roots are equal (one root of multiplicity 3 or two distinct real roots, one of which is a double root).

**Discriminant Calculator of a quartic polynomial**

Consider the quartic polynomial ax⁴ + bx³ + cx² + dx + e. The formula for its discriminant reads:

256a³e³ – 192a²bde² – 128a²c²e² + 144a²cd²e – 27a²d⁴ + 144ab²ce² – 6ab²d²e – 80abc²de + 18abcd³ + 16ac⁴e – 4ac³d² – 27b⁴e² + 18b³cde – 4b³d³ – 4b²c³e + b²c²d².

If there are four separate real roots or four distinct non-real roots, D is higher than 0. (two pairs of conjugate complex roots).

If there are two separate real roots and two distinct non-real roots, D is less than 0. (one pair of conjugate complex roots).

D = 0 if and only if two or more equal roots exist. There are six options available:

- Three separate real roots, one of which is double.
- Two unique real roots that are both double.
- Then, two unique real roots, one of which has three multiplicities.
- One real root with four multiplicities.
- Similarly, one real double root and two non-real complex conjugate roots.
- Then, two double non-real complex conjugate roots.

**Discriminant Calculator of a quintic polynomial**

We don’t give the formula as it is, with 59 terms, each of which is an eight-degree monomial in six variables.

You might be wondering how to find a quintic polynomial’s discriminant.

Fortunately, we have a discriminant calculator that includes this formula. When you need to consider a quintic polynomial, this is the tool to use.

After that, use the following set of rules to figure out what your polynomial’s properties are:

If there are five unique real roots or one real root and two pairs of non-real complex conjugate roots, D is greater than 0.

If there are three unique real roots and one pair of non-real complex conjugate roots, D is less than 0.

D = 0 if and only if two or more equal roots exist. There are four options:

- Six distinct situations with only genuine roots.
- One real root of multiplicity three and one pair of non-real complex conjugate roots.
- Then, one real single root and one pair of non-real complex conjugate double roots
- Similarly, one real single root and one pair of non-real complex conjugate double roots

**Discriminant Calculator Standard Form**

The standard discriminant form for the quadratic equation ax² + bx + c = 0 is

Discriminant, D = b² – 4ac

Where

- a is the coefficient of x².
- b is the coefficient of x.
- c is a constant term.

**How to Use The Discriminant Calculator**

Follow the steps below to utilise the discriminant calculator:

- Choose the degree of the polynomial you wish to consider first. Choose polynomials with degrees ranging from 2 to 5, such as quadratic (degree 2), cubic (degree 3), quartic (degree 4) or quintic (degree 5) polynomials.
- If you need to find the discriminant of a quadratic equation, for example, use second as the degree.
- Input all of your polynomial’s coefficients, including those that are zero.
- Enjoy the outcome, which our discriminant calculator provides right away!

**How to Find The Discriminant Using Discriminant Calculator?**

The discriminant calculator tells you how to solve the provided equation problems step by step. Whether you need to solve quadratic equations or higher degree polynomial equations, this calculator can help! Follow these inputs to obtain the discriminant of the equations:

**Inputs:**

To begin, choose the degree of the polynomial you want to get the discriminant for from the dropdown menu of this tool.

Then, for the selected equation, input the coefficient values. (Enter the values based on the polynomial degree selected)

Finally, press the computer key.

**Outputs:**

The discriminant calculator will reveal the following:

Then, the provided equation’s discriminant.

Then, the roots’ natural state.

Complete the discriminant computation.

**Discriminant Calculator of a quadratic equation**

discriminant of a quadratic equation

The discriminant of the quadratic equation following ax²+bx+c=0 is equal to b²−4ac. The notation used for the discriminant is Δ (delta), so we have Δ=b²−4ac.

The calculator has a feature that allows you to calculate the discriminant of quadratic equations online. To find the equation’s discriminant, do the following: 3x²+4x+3=0, enter discriminant(3⋅x²+4⋅x+3=0;x), the calculator returns the result -20.

Calculating a polynomial’s discriminant allows you to figure out how many roots a quadratic equation has:

When the discriminant calculation yields a negative value, the equation has no root.

However, when the discriminant yields zero, the equation has a root.

When the discriminant yields a positive number, the equation has two unique roots.

However, the equation solver can be used to find the roots of an equation.

**Discriminant Calculator Examples**

### Example 1

Find the discriminant of the following equation: √3×2 + 10x − 8√3 = 0.

Solution:

The given quadratic equation is √3×2 + 10x − 8√3 = 0. Comparing this with ax² + bx + c = 0, we get a = √3, b = 10, and c = -8√3.

The quadratic discriminant formula is:

D = b² – 4ac

= (10)2 – 4(√3)(-8√3)

= 100 + 96

= 196

Answer: The discriminant = 196.

### Example 2

What is the discriminant of quadratic equation 9z² − 6b²z − (a⁴ − b⁴) = 0.

Solution:

By comparing the given equation with ax² + bx + c = 0, we get a = 9, b = -6b², and c = – (a⁴ − b⁴). Its discriminant is,

D = b² – 4ac

= (-6b²)² – 4 (9) [-(a⁴ − b⁴)]

= 36b⁴ + 36a⁴ – 36b⁴

= 36a⁴

Answer: The discriminant of the given quadratic equation is 36a4.

**Discriminant Calculator Multivariable**

Determine whether the quadratic equations below have two real roots, one real root, or no real roots. (a) 3x² − 5x − 7 = 0 (b) 2x² + 3x + 3=0.

Solution:

(a) By comparing the given equation with ax² + bx + c = 0, we get a = 3, b = -5, and c = -7. Its discriminant is,

D = b² – 4ac

= (-5)2 – 4(3)(-7)

= 25 + 84

= 109

As a result, we have a positive discriminant, and the equation has two real roots.

(b) By comparing the given equation with ax² + bx + c = 0, we get a = 2, b = 3, and c = 3. Its discriminant is,

D = b² – 4ac

= (3)² – 4(2)(3)

= 9 – 24

= -15

As a result, we get a negative discriminant, and the equation has no real roots.

Answer: (a) Two real roots (b) No real roots.

**Discriminant Calculator with k**

Find the values of k

Step 1 of 2 : Write down the given Quadratic equation. The given Quadratic equation is. kx² – 14x + 8 = 0.

Then, step 2 :

Find the value of k. Here it is given that one root of the quadratic equation kx² – 14x + 8 = 0 is 2. So by the given condition. Hence the required value of k = 5.

**FAQs On Discriminant Calculator**

### What is Discriminant Meaning?

In math, the discriminant is a function of polynomial coefficients and is defined for polynomials. It describes the character of roots or, to put it another way, it discriminates against them. The discriminant of a quadratic equation, for example, is used to find:

- What is the number of roots on it?
- Whether or if the roots are real?

### What is the Discriminant Formula?

Different polynomials have different discriminant formulas:

OR D = b² – 4ac is the discriminant of the quadratic equation ax² + bx + c = 0.

D = b²c² – 4ac³ – 4b³d – 27a²d² + 18abcd is the discriminant of a cubic equation ax³ + bx² + cx + d = 0.

### How to Calculate the Discriminant of a Quadratic Equation?

To find the discriminant of a quadratic equation, do the following:

Compare the preceding equation to ax² + bx + c = 0 to find a, b, and c.

Substitute the values for D = b²- 4ac in the discriminant formula.

### What if Discriminant = 0?

If the discriminant of a quadratic equation ax² + bx + c = 0 is 0 (i.e., if b² – 4ac = 0), then the quadratic formula becomes x = -b/2a and hence the quadratic equation has only one real root.

### What Does a Positive Discriminant Tell Us?

If the discriminant of a quadratic equation ax² + bx + c = 0 is positive (i.e., b² – 4ac is greater than 0), the quadratic formula becomes x = (-b ± √(positive number)) / 2a, with only two real and different roots.

### What Does Negative Discriminant Tell Us?

If the discriminant of a quadratic equation ax² + bx + c = 0 is negative (i.e., b² – 4ac is less than 0), the quadratic formula becomes x = (-b ± √(negative number)) / 2a, with just two complex and separate roots.

### What is the Formula for Discriminants of Cubic Equations?

The discriminant of a cubic equation is in terms of its coefficients, which is given by the formula ax³ + bx² + cx + d = 0 or, D = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd.

### When the Discriminant is Zero, How Many Numbers of Solution Would a Quadratic Equation Have?

A component of the square root of the quadratic formula, b²-4ac, is the discriminant or determinant of a quadratic equation. There must be a unique solution if the discriminant is equal to zero. There is no solution if the discriminant is less than zero. If it is greater than zero, there are two possible real solutions to the equation.

### What are the Various Forms of a Quadratic Equation?

A quadratic equation can be written in three different ways.

Vertex form is a factored form.

In this situation, the usual form of a quadratic equation is y = ax² + bx + c, and the discriminant of a quadratic equation is b²- 4ac. y = (axe + c) (bx + d) is the factored form of the quadratic equation. y = a (x + b)² + c is the vertex version of the quadratic equation.

The a, b, and c numbers must be noted.

### What is the Significance of Quadratic Equations?

In our daily lives, quadratic equations are used to calculate areas, determine the profit of a product, and calculate the speed of an object, among other things. Within the quadratic equation, there will be at least one squared variable, which is written as ax² + bx + c = 0, where x is the variable, a, b, and c are constants, and a does not equal zero. To determine the answer to the problem, utilise the discriminant quadratic function.

### What does a discriminant of 0 mean?

If the discriminant is 0, it means you have a 0 under the square root in the quadratic formula. The square root of zero is 0. The plus and minus parts of the quadratic formula simply vanish when this happens. Only one option remains.