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 5x² + 3x + 2 = 0. Also, determine the nature of its roots.

Solution:

The given quadratic equation is 5x² + 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 2x² + 8x + 8 = 0. Also, determine the nature of its roots.

Solution:

The given quadratic equation is 2x² + 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 - 4x² + 6x - 4 = 0.

Solution:

The given cubic equation is x³ - 4x² + 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.