A quadratic equation will have a solution based on the value of its discriminant. The term inside a radical symbol (square root) of a quadratic formula is said to be a discriminant. The discriminant in Math is used to find the nature of the roots of a quadratic equation. The value of the discriminant will determine if the roots of the quadratic equation are real or imaginary, equal or unequal.

Discriminant Definition in Math

The discriminant of a polynomial is a function of its coefficients which gives an idea about the nature of its roots. For a quadratic polynomial ax² + bx + c, the formula of discriminant is given by the following equation :

D = b² – 4ac

For a cubic polynomial ax³ + bx² + cx + d, its discriminant is expressed by the following formula

D= b²c²−4ac³−4b³d−27a²d²+18abcd

Similarly, for polynomials of higher degrees also, the discriminant is always a polynomial function of the coefficients. For higher degree polynomials, the discriminant equation is significantly large. The number of terms in discriminant exponentially increases with the degree of the polynomial. For a fourth-degree polynomial, the discriminant has 16 terms; for fifth-degree polynomial, it has 59 terms, and for a sixth-degree polynomial, there are 246 terms.

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Discriminant Formula

In algebra, the quadratic equation is expressed as ax² + bx + c = 0, and the quadratic formula is represented as \(x =\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\).

Therefore, the discriminant formula for the general quadratic equation is

Discriminant, D = b² – 4ac

Where

a is the coefficient of x²

b is the coefficient of x

c is a constant term

Discriminant of a Polynomial

The discriminant of a quadratic polynomial is the portion of the quadratic formula under the square root symbol: b²-4ac, that tells whether there are two solutions, one solution, or no solutions to the given equation.

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

Relationship Between Discriminant and Nature of Roots

The discriminant value helps to determine the nature of the roots of the quadratic equation. The relationship between the discriminant value and the nature of roots are as follows:

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

Discriminant Example

Example 1: Determine the discriminant value and the nature of the roots for the given quadratic equation 3x²+2x+5.

Solution:

Given: The quadratic equation is 3x²+2x+5

Here, the coefficients are:

a = 3

b = 2

c = 5

The formula to find the discriminant value is D = b² – 4ac

Now, substitute the values in the formula

Discriminant, D = 2² – 4(3)(5)

D = 4 – 4 (15)

D = 4 – 60

D = -56

The discriminant value is -56, which is less than 60.

I.e., -56 < 0

Therefore, the roots are not real.

Hence, the quadratic equation has no real roots.

Example 2: Determine the discriminant value and the nature of the roots for the given quadratic equation 2x²+8x+8.

Solution:

Given: The quadratic equation is 2x²+8x+8

Here, the coefficients are:

a = 2, b = 8 and c = 8

The formula to find the discriminant value is D = b² – 4ac

Now, substitute the values in the formula

Discriminant, D = 8² – 4(2)(8)

D = 64 – 4 (16)

D = 64 – 64

D = 0

The discriminant value is 0

I.e., D = 0

Therefore, the roots are real and equal

Hence, the quadratic equation has a double root (repeated roots)

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