Quadratic Equations in one variable

Chapter 1: Quadratic Equations in One Variable

Overview

The chapter “Quadratic Equations in One Variable” introduces students to a vital branch of algebra that forms the foundation for solving real-life mathematical problems and higher-level concepts in mathematics. A quadratic equation is a polynomial equation of degree two, containing a single variable. This chapter builds the analytical and problem-solving skills of students by helping them recognize, solve, and interpret quadratic equations systematically.

Mastering this chapter is essential not only for board examinations but also for laying the groundwork for future studies in algebra, calculus, and other branches of mathematics.

What is a Quadratic Equation in One Variable?

A quadratic equation in one variable is an equation that can be written in the standard form:
ax² + bx + c = 0,
where a ≠ 0, and a, b, c are real numbers.

Here,

x is the variable,
a is the coefficient of x² (quadratic term),
b is the coefficient of x (linear term), and
c is the constant term.

Example:
x² – 5x + 6 = 0 is a quadratic equation.
It is in the standard form where a = 1, b = -5, and c = 6.

Nature and Types of Roots

The roots (or solutions) of a quadratic equation are the values of x that satisfy the equation. The nature of the roots is determined by the discriminant (D), which is calculated as:
D = b² – 4ac

Based on the value of D, the nature of the roots varies:

If D > 0: Two distinct real roots
If D = 0: Two equal real roots
If D < 0: Two non-real complex roots
Example:
For the equation x² – 4x + 3 = 0,
Here, a = 1, b = -4, c = 3
D = (-4)² – 4×1×3 = 16 – 12 = 4 > 0 ⇒ Two distinct real roots.

Methods of Solving Quadratic Equations

This chapter outlines three primary methods to solve quadratic equations in one variable:

1. Factorization Method

In this method, the quadratic expression is factorized into two linear factors. This method is effective when the equation can be easily factorized.

Example:
x² – 7x + 10 = 0
⇒ (x – 2)(x – 5) = 0
⇒ x = 2 or x = 5

2. Completing the Square Method

This approach involves transforming the equation into a perfect square trinomial and then taking the square root of both sides.

Example:
x² + 6x + 5 = 0
⇒ x² + 6x = -5
⇒ x² + 6x + 9 = 4 (add 9 to both sides)
⇒ (x + 3)² = 4
⇒ x + 3 = ±2
⇒ x = -1 or -5

3. Quadratic Formula Method

The quadratic formula provides a direct solution and is applicable to all types of quadratic equations:

x = (-b ± √(b² – 4ac)) / 2a

Example:
For x² – 3x + 2 = 0
a = 1, b = -3, c = 2
x = [3 ± √((-3)² – 4×1×2)] / (2×1)
= [3 ± √(9 – 8)] / 2
= [3 ± 1] / 2
⇒ x = 2 or x = 1

Applications of Quadratic Equations

Though the focus of this chapter is on algebraic solutions, quadratic equations also model numerous real-world situations like:

Calculating area and dimensions of geometric figures
Determining the time and speed in motion-related problems
Solving age-based word problems
Profit and loss analysis in basic commercial mathematics

The syllabus restricts applications to simple, logical word problems that can be translated into quadratic equations.

Key Concepts to Remember

The coefficient a must never be zero in a quadratic equation.
Every quadratic equation has two roots, which may be real or complex.
Use the discriminant (D = b² – 4ac) to determine the nature of the roots.
All three methods (factorization, completing the square, and formula method) lead to the same roots if applied correctly.
Quadratic equations are symmetric in nature due to the presence of the square term.