Discriminant: Formula, Nature of Roots and its Calculation

Discriminant is a part of quadratic equations and in mathematics it helps to understand the number of real solutions present in a quadratic equation. The expression used to find the discriminant is the expression located under the radical in the quadratic formula. In the case of a quadratic equation ax2 + bx + c = 0, the discriminant is b2 − 4ac; for a cubic equation x3 + ax2 + bx + c = 0, the discriminant is a2b2 + 18abc − 4b3 − 4a3c − 27c2.

Key Takeaways- Discriminant, Polynomials, Discriminant Formula, Coefficients, Quadratic equation, Nature of roots

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What is Discriminant in Math?

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The polynomial discriminant is a function of its coefficients and gives an idea of âÂÂ\(\Box\)ÂÂ\(\Box\)âÂÂ\(\Box\)ÂÂ\(\Box\)its root properties. For the  quadratic polynomial ax² + bx + c, the  discriminant is given by: D = b² – 4ac.

For cubic polynomials ax³ + bx² + cx + d, the  discriminant is given by: 

D= b²c²−4ac³−4b³d−27a²d²+18abcd.

Similarly, a discriminant is always a polynomial function of the coefficients of a higher degree polynomial. For higher-order polynomials, the discriminant equation is very large. The number of discriminant terms increases exponentially with the degree of the polynomial. For a fourth-order polynomial, the discriminant has 16 terms. There are 59 terms for a fifth-order polynomial and 246 terms for a sixth-order polynomial.

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Discriminant Formula

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The polynomial discriminant is a function of its coefficients and is represented by the uppercase "D" or the delta symbol Δ. It shows the root nature of any quadratic equation with rational coefficients a, b, and c. The quadratic equation can simply give the number of real roots or x-intercepts. This formula is used to determine if the root of a quadratic equation is real or imaginary.

We know b²–4ac determines whether the quadratic equation ax²+bx+c=0 has real roots or not, b²–4ac is known as the discriminant of these quadratic equations.

Therefore, a quadratic equation ax²+bx+c=0 has

Two distinct real roots, if b²–4ac>0

Two equal real roots, if b²–4ac=0

No real roots, if b²–4ac<0

Discriminants Calculation

To find a discriminant for a cubic or quadratic equation, you need to compare the given equation with its canonical form and first determine the coefficients. Then substitute the coefficients of the relevant expression to find the discriminant.

The discriminant of a quadratic equation ax² + bx + c = 0, when they exist, in terms of the coefficients a,b,c. The solutions are-

 x = −b ± √b²−4ac / 2a ,  x = −b - √b²−4ac / 2a 

provided that b²−4ac≥0

The quantity b²−4ac is called the discriminant of the quadratic, often denoted by Δ, and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:

• Δ>0 tells us the equation has two distinct real roots
• Δ=0 tells us the equation has one (repeated) real root
• Δ<0 tells us the equation has no real roots.

Here, the expression that is inside the square root of the quadratic formula is called the discriminant of the quadratic equation.

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Discriminant and nature of roots

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The category by which the roots fall simply refers to the nature of the roots. The roots can be made real, unequal, or even equal. The roots will be fictitious if the object of distinction is negative. 

Calculate the discriminant value of a cubic equation to discover the nature of its integer solution. A cubic equation has an integer solution if the

discriminant is 0 and if all the coefficients of the cubic equation are real.

Relationship Between Discriminant and Nature of Roots

The relationship between the discriminant value and the nature of roots are as follows:

• If discriminant > 0, the roots are real and unequal
• If discriminant = 0, the roots are real and equal
• If discriminant < 0, the roots are not real (we get a complex solution)

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Discriminant in Polynomials

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The discriminant of a polynomial is an invariant which determines whether or not a polynomial has repeated roots.

The discriminant is a homogeneous polynomial in the coefficients. It is quasi-homogeneous in the coefficients since it is also a homogeneous polynomial in the roots. The discriminant of a polynomial of degree n is homogeneous of degree 2n − 2 in the coefficients.

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Things to Remember 

• Correct placement of the equations is essential. Otherwise, you will not be able to get a solution.
• Keep an eye out for negative b². Since it cannot be negative, make sure to change it to positive. The square of either positive or negative will always be positive.
• Ensure that 2a and the square root of the entire (b² − 4ac) is placed at the denominator.
• While using a calculator, the number needs to be rounded on a particular number of decimal places.
• Retain the +/−. Watch out for two solutions.

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