Roots of Quadratic Equation

Quadratic equations are one of the most important as well as fundamental topics in algebra, and their roots play a crucial role in understanding their behaviour. In this article, we will explore the concept of quadratic equations, different methods to find their roots, properties of roots, and real-world applications.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two of the form:

𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0

where:

𝑎, 𝑏, and 𝑐 are coefficients, with 𝑎 ≠ 0,

𝑥 is the variable.

The solutions to the quadratic equation are called roots. Roots can be real or complex numbers and represent the values of 𝑥, satisfying that equation.

Methods to find the roots of a quadratic equation:

There are several methods to find the roots of a quadratic equation. Below, we discuss the most common ones:

1. Quadratic Formula

The quadratic formula is the most straightforward and widely used method to find the roots of a quadratic equation. The formula is:

𝑥 = (−𝑏 ± √(𝑏² − 4𝑎𝑐)) / 2𝑎

Here, 𝑏² − 4𝑎𝑐 is called the discriminant, and it determines the nature of the roots:

• If 𝑏² − 4𝑎𝑐 > 0, the equation has two distinct real roots.

• If 𝑏² − 4𝑎𝑐 = 0, the equation has one real root (a repeated root).

• If 𝑏² − 4𝑎𝑐 < 0, the equation has two complex roots.

Example:

Solve the quadratic equation 2𝑥² + 5𝑥 − 3 = 0.

Using the quadratic formula:

𝑥 = (−5 ± √(5² − 4×2×−3)) / 2×2

𝑥 = (−5 ± √(25 + 24)) / 4

𝑥 = (−5 ± √49) / 4

𝑥 = (−5 ± 7) / 4

Thus, the roots are:

𝑥₁ = (−5 + 7) / 4 = 2 / 4 = 0.5

𝑥₂ = (−5 − 7) / 4 = −12 / 4 = −3

2. Factorisation Method

If the quadratic equation can be factored into two binomials, the roots can be found by setting each factor equal to zero.

Example:

Solve the quadratic equation 𝑥² − 5𝑥 + 6 = 0.

Factor the equation:

(𝑥 − 2)(𝑥 − 3) = 0

Set each factor equal to zero:

𝑥 − 2 = 0 ⇒ 𝑥 = 2

𝑥 − 3 = 0 ⇒ 𝑥 = 3

Thus, the roots are 𝑥 = 2 and 𝑥 = 3.

3. Completing the Square (Sridharacharya Formula)

Completing the square is another method used to solve quadratic equations. It involves rewriting the equation in the form (𝑥 + 𝑝)² = 𝑞 and then solving for 𝑥.

Example:

Solve the quadratic equation 𝑥² + 6𝑥 + 5 = 0.

Step 1: Move the constant term to the other side:

𝑥² + 6𝑥 = −5

Step 2: Complete the square:

𝑥² + 6𝑥 + 9 = −5 + 9

(𝑥 + 3)² = 4

Step 3: Solve for 𝑥:

𝑥 + 3 = ±2

𝑥 = −3 ± 2

Thus, the roots are:

𝑥₁ = −3 + 2 = −1

𝑥₂ = −3 − 2 = −5

Properties of the Roots of a Quadratic Equation:

The roots of a quadratic equation 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 have the following relationship:

1. Sum of the Roots: Sum = −𝑏 / 𝑎

2. Product of the Roots: Product = 𝑐 / 𝑎

Applications of Quadratic Equations:

Quadratic equations have numerous real-life applications, some are discussed below:

1. In physics: Calculating projectile motion, trajectories, and energy.

2. In engineering: Designing structures, analysing forces, and optimizing systems.

3. In economics: Modelling profit, cost, and revenue functions.

4. In geometry: Solving problems related to areas and dimensions.

5. In computer graphics: Understanding curves and shapes.