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 = ${\left(-5\right)}^{2}$ - 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 (non-real) 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. Real-World 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 = ${\left(-6\right)}^{2}$ - 4(1)(9) = 0.

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