# 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 ax^{2} + bx + c = 0 is, Δ (or) D = b^{2} - 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 ax^{2} + bx + c = 0. According to the quadratic formula, the roots can be found using x = [-b ± √ (b^{2} - 4ac) ] / [2a]. Here, b^{2} - 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 ax^{3} + bx^{2} + cx + d = 0 is, Δ (or) D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 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 5x^{2 }+ 3x + 2 = 0. Also, determine the nature of its roots.

**Solution:**

The given quadratic equation is 5x^{2 }+ 3x + 2 = 0.

Comparing this with ax^{2} + bx + c = 0, we get a = 5, b = 3, and c = 2.

Using discriminant formula,

D = b^{2 }- 4ac

= 3^{2 }- 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 2x^{2 }+ 8x + 8 = 0. Also, determine the nature of its roots.

**Solution:**

The given quadratic equation is 2x^{2 }+ 8x + 8 = 0.

Comparing this with ax^{2} + bx + c = 0, we get a = 2, b = 8, and c = 8.

Using the discriminant formula,

D = b^{2 }- 4ac

= 8^{2}- 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 x^{3} - 4x^{2} + 6x - 4 = 0.

**Solution:**

The given cubic equation is x^{3} - 4x^{2} + 6x - 4 = 0.

Comparing this with ax^{3} + bx^{2} + cx + d = 0, we get a = 1, b = -4, c = 6, and d = -4.

Using the discriminant formula,

D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 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 ax
^{2}+ bx + c = 0 is, Δ (or) D = b^{2}- 4ac. - The discriminant formula of a cubic equation ax
^{3}+ bx^{2}+ cx + d = 0 is, Δ (or) D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd.

### How To Derive the Discriminant Formula of a Quadratic Equation?

Let us derive the discriminant formula of a quadratic equation ax^{2} + bx + c = 0. By quadratic formula, the solutions of this equation are found using x = [-b ± √ (b^{2} - 4ac) ] / [2a]. Here b^{2} - 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 b^{2} - 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 ax^{2} + bx + c = 0 is D = b^{2} - 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 ax^{3} + bx^{2} + cx + d = 0 is denoted by Δ (or) D and is found using the formula Δ (or) D = b^{2}c^{2 }− 4ac^{3 }− 4b^{3}d − 27a^{2}d^{2 }+ 18abcd.

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