Can a quadratic equation have three roots?

No. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n complex roots. A quadratic equation is a second-degree polynomial (n = 2) and thus can have at most two roots.

What are imaginary roots?

When the discriminant D = b² - 4ac is negative, the square root term contains a negative value, yielding roots with the imaginary unit i (where i = √-1). These are complex numbers representing points where the parabola does not cross the x-axis.

Back to Class IX Mathematics

Class IX Mathematics

Chapter 4: Linear Equations in Two Variables

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

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Syllabus Sections

Chapter Overview

Welcome to Class IX Mathematics: Linear Equations in Two Variables. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Linear Equations in One VariableCoordinate Geometry

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Standard form ax + by + c = 0

Concept Explanation

An equation of the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero, is a linear equation in two variables x and y. It represents a straight line when plotted on a graph.

Mathematical Representation

ax + by + c = 0 \quad (a^2 + b^2 \neq 0)

Study Guideline: Convert any linear equation into this standard form by moving all terms to one side, keeping the coefficient of x positive where possible.

2Solutions of a linear equation

Concept Explanation

A linear equation in two variables has infinitely many solutions. A solution is a pair of values (x, y) that satisfies the equation. When substituted, these values make the left-hand side equal to the right-hand side.

Mathematical Representation

a(x_0) + b(y_0) + c = 0 \implies (x_0, y_0) \text{ is a solution}

Study Guideline: To find solutions, substitute an arbitrary value for x and solve the equation for y, or vice versa.

3Graph of a linear equation in two variables

Concept Explanation

The graph of a linear equation in two variables is a straight line. Every point on the line is a solution of the equation, and every solution of the equation lies on this line.

Mathematical Representation

y = mx + c \quad \text{where } m \text{ is slope and } c \text{ is y-intercept}

Study Guideline: To plot the line, find at least two distinct solutions (typically the x-intercept and y-intercept), plot them, and draw a straight line through them.

4Equations of lines parallel to axes

Concept Explanation

Equations of lines parallel to the coordinate axes are simple. An equation of the form x = k represents a vertical line parallel to the y-axis, and an equation of the form y = k represents a horizontal line parallel to the x-axis.

Mathematical Representation

x = k \, \parallel \text{ y-axis}, \quad y = k \, \parallel \text{ x-axis}

Study Guideline: The equation x = 0 represents the y-axis itself, and y = 0 represents the x-axis itself.