Nature of Roots
Nature of Roots of a Quadratic Equation
Quadratic equations are of the form:
a${x}^{2}$ + bx + c = 0
Where:
 a, b, and c are coefficients.
 x is the variable.
The "nature of roots" refers to the characteristics of the solutions (roots) of a quadratic equation. It is determined by the discriminant, D, which is calculated as D = ${b}^{2}$  4ac.
1. Discriminant (D):
 The discriminant is a crucial factor in determining the nature of the roots of a quadratic equation.
2. Discriminant and Nature of Roots:
a. D > 0 (Positive Discriminant):  If the discriminant is positive, the quadratic equation has two distinct real roots.
 These real roots are often referred to as "unequal" or "distinct" roots.
 Example: 2${x}^{2}$  5x + 2 = 0 has two real and distinct roots.

The discriminant (D) is calculated as D = ${b}^{2}$  4ac for the given equation.
In this case:
 a = 2
 b = 5
 c = 2
D = ${(5)}^{2}$  4(2)(2) = 25  16 = 9
 Since the discriminant is positive (D > 0), this quadratic equation has two distinct real roots.

To find the roots, you can use the quadratic formula: x = (b ± √D) / (2a)
x₁ = ((5) + √9) / (2 * 2) = (5 + 3) / 4 = 8/4 = 2
x₂ = ((5)  √9) / (2 * 2) = (5  3) / 4 = 2/4 = 1/2
So, the roots of the equation 2${x}^{2}$  5x + 2 = 0 are x₁ = 2 and x₂ = 1/2. These are two distinct real roots, as indicated by the positive discriminant.
b. D = 0 (Zero Discriminant):  If the discriminant is zero, the quadratic equation has one real root (a repeated root).
 This means that both solutions are identical.
 Example: ${x}^{2}$  4x + 4 = 0 has one real and repeated root: x = 2 (twice).
c. D < 0 (Negative Discriminant):  If the discriminant is negative, the quadratic equation has two complex (nonreal) roots.
 These roots are often referred to as "imaginary" or "complex" roots.
 Example: ${x}^{2}$ + 2x + 5 = 0 has two complex roots.
3. Use of the Discriminant:

The discriminant helps in understanding the nature of solutions to a quadratic equation.

It can be used to determine whether a quadratic equation has real roots, repeated roots, or complex roots.

It provides insights into the behavior of the equation's solutions.
4. RealWorld Applications:

The concept of the nature of roots is essential in various fields, including physics, engineering, economics, and computer science.

In physics, it can describe the motion of projectiles, such as when a ball is thrown into the air.

In engineering, it can help design systems with specific behaviors, such as electrical circuits.

In economics, it can model revenue and profit functions.
Example:

Consider the quadratic equation: ${x}^{2}$  6x + 9 = 0.

Calculate the discriminant: D = ${(6)}^{2}$  4(1)(9) = 0.

Since D = 0, this equation has one real and repeated root, which is x = 3.