CBSE Class 10 Mathematics Chapter 4 - Quadratic Equations - Introduction

CBSE Class 10 Mathematics  Chapter 4 - Quadratic Equations - Introduction

Quadratic Equations

The standard form of a quadratic equation

A quadratic equation is a second-degree polynomial equation in a single variable (most commonly written as 'x'). It has the general form:

ax² + bx + c = 0

In this equation:

• a,  b and c are constants, with a ≠ 0.
• 'x' represents the variable we're trying to solve for.
• The highest power of 'x' in the equation is 2 which makes it a quadratic equation.
• This is called the standard form of a quadratic equation.
• A quadratic polynomial becomes a quadratic equation when it is set equal to zero.
• The values of 'x' that satisfy the quadratic equation is referred to as the roots of the quadratic equation.

Defining Roots of a Quadratic Equation

• A real number 'α' is considered a root of a quadratic equation ax² + bx + c = 0, where a ≠ 0, if when we replace 'x' with 'α' on the left-hand side(L.H.S.), the equation becomes true (0 = 0).
• In other words, a real number 'α' is a root or solution of the quadratic equation if aα² + bα + c = 0.

Relation to Quadratic Polynomials

• The roots of a quadratic equation are closely related to the zeros of the corresponding quadratic polynomial. The terms "roots" and "zeros" are used interchangeably in this context.

Number of Roots

• A quadratic polynomial can have at most two distinct zeros. Therefore, any quadratic equation can have at most two real roots.

Vertex Form of quadratic equation

• A quadratic equation can also be expressed in vertex form, which is useful for graphing:

y = a(x - h)² + k

Here, (h, k) represents the vertex of the parabola.