What are Discriminant Formulas? Examples

from a handpicked tutor in LIVE 1-to-1 classes

Discriminant Formulas

The discriminant formulas are used to find the discriminant of a polynomial equation. Especially, the discriminant of a quadratic equation is used to determine the number and the nature of the roots. The discriminant of a polynomial is a function that is made up of the coefficients of the polynomial. Let us learn the discriminant formulas along with a few solved examples.

What Is Discriminant Formulas?

The discriminant formulas give us an overview of the nature of the roots. The discriminant of a quadratic equation is derived from the quadratic formula. The discriminant is denoted by D or Δ. The discriminant formulas for a quadratic equation and cubic equation are:

Discriminant Formula of a Quadratic Equation

The discriminant formula of a quadratic equation ax2 + bx + c = 0 is, Δ (or) D = b2 – 4ac. We know that a quadratic equation has a maximum of 2 roots as its degree is 2. We know that the quadratic formula is used to find the roots of a quadratic equation ax2 + bx + c = 0. According to the quadratic formula, the roots can be found using x = [-b ± √ (b2 – 4ac) ] / [2a]. Here, b2 – 4ac is the discriminant D and it is inside the square root. Thus, the quadratic formula becomes x = [-b ± √D] / [2a]. Here D can be either > 0, = 0, (or) < 0. Let us determine the nature of the roots in each of these cases.

• If D > 0, then the quadratic formula becomes x = [-b ± √(positive number)] / [2a] and hence in this case the quadratic equation has two distinct real roots.
• If D = 0, then the quadratic formula becomes x = [-b] / [2a] and hence in this case the quadratic equation has only one real root.
• If D < 0, then the quadratic formula becomes x = [-b ± √(negative number)] / [2a] and hence in this case the quadratic equation has two distinct complex roots (this is because the square root of a negative number results in an imaginary number. For example, √(-4) = 2i).

Discriminant Formula of a Cubic Equation

The discriminant formula of a cubic equation ax3 + bx2 + cx + d = 0 is, Δ (or) D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd. We know that a cubic equation has a maximum of 3 roots as its degree is 3. Here,

• If D > 0, all the three roots are real and distinct.
• If D = 0, then all the three roots are real where at least two of them are equal to each other.
• If D < 0, then two of its roots are complex numbers and the third root is real.

We can see the applications of the discriminant formulas in the following section.

Examples Using Discriminant Formulas

Example 1: Determine the discriminant of the quadratic equation 5×2 + 3x + 2 = 0. Also, determine the nature of its roots.

Solution:

The given quadratic equation is 5×2 + 3x + 2 = 0.

Comparing this with ax2 + bx + c = 0, we get a = 5, b = 3, and c = 2.

Using discriminant formula,

D = b2 – 4ac

= 32 – 4(5)(2)

= 9 – 40

= -31

Answer: The discriminant is -31. This is a negative number and hence the given quadratic equation has two complex roots.

Example 2: Determine the discriminant of the quadratic equation 2×2 + 8x + 8 = 0. Also, determine the nature of its roots.

Solution:

The given quadratic equation is 2×2 + 8x + 8 = 0.

Comparing this with ax2 + bx + c = 0, we get a = 2, b = 8, and c = 8.

Using the discriminant formula,

D = b2 – 4ac

= 82- 4(2)(8)

= 64 – 64

= 0

Answer: The discriminant is 0 and hence the given quadratic equation has two complex roots.

Example 3: Determine the nature of the roots of the cubic equation x3 – 4×2 + 6x – 4 = 0.

Solution:

The given cubic equation is x3 – 4×2 + 6x – 4 = 0.

Comparing this with ax3 + bx2 + cx + d = 0, we get a = 1, b = -4, c = 6, and d = -4.

Using the discriminant formula,

D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd

= (-4)2(6)2 − 4(1)(6)3 − 4(-4)3(-4) − 27(1)2(-4)2 + 18(1)(-4)(6)(-4)

= -16

Answer: Since the discriminant is a negative number, the given cubic equation has two complex roots and one real root.

FAQs on Discriminant Formulas

What Are Discriminant Formulas?

The discriminant of a polynomial equation is a function which is in terms of its coefficients. The discriminant of an equation is used to determine the nature of its roots. The discriminant formulas are as follows:

• The discriminant formula of a quadratic equation ax2 + bx + c = 0 is, Δ (or) D = b2 – 4ac.
• The discriminant formula of a cubic equation ax3 + bx2 + cx + d = 0 is, Δ (or) D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd.

How To Derive the Discriminant Formula of a Quadratic Equation?

Let us derive the discriminant formula of a quadratic equation ax2 + bx + c = 0. By quadratic formula, the solutions of this equation are found using x = [-b ± √ (b2 – 4ac) ] / [2a]. Here b2 – 4ac is inside the square root and hence we can determine the nature of the roots by using the properties of the square root (such as the square root of a positive number is a real number, the square root of a negative number is an imaginary number, and the square root of 0 is 0). Thus, the discriminant of the quadratic equation is b2 – 4ac.

What Are the Applications of the Discriminant Formula?

The discriminant formula is used to determine the nature of the roots of a quadratic equation. The discriminant of a quadratic equation ax2 + bx + c = 0 is D = b2 – 4ac.

• If D > 0, then the equation has two real distinct roots.
• If D = 0, then the equation has only one real root.
• If D < 0, then the equation has two distinct complex roots.

What Is the Discriminant Formula of a Cubic equation?

The discriminant formula of a cubic equation ax3 + bx2 + cx + d = 0 is denoted by Δ (or) D and is found using the formula Δ (or) D = b2c2 − 4ac3 − 4b3d − 27a2d2 + 18abcd.

visual curriculum