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.

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Class X Mathematics

Chapter 4: Quadratic Equations

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 X Mathematics: Quadratic Equations. 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

Polynomial expressions factorizationLinear equations balancing

Detailed Subtopics Study Guide

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

1Standard form ax² + bx + c = 0

Concept Explanation

A quadratic equation is a second-degree polynomial equation. Its standard form is ax² + bx + c = 0, where a, b, and c are real numbers and the leading coefficient a is not zero.

Mathematical Representation

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

Study Guideline: Ensure all terms are on one side of the equation and arranged in descending order of power before identifying a, b, and c.

2Roots by factorization

Concept Explanation

Solving a quadratic equation by factorization involves splitting the middle term (linear coefficient b) into two parts that multiply to ac and add to b. Factoring the equation allows us to apply the Zero Product Property.

Mathematical Representation

ax^2 + bx + c = 0 \implies (px + q)(rx + s) = 0 \implies x = -\frac{q}{p} \, \text{ or } \, x = -\frac{s}{r}

Study Guideline: Find two numbers whose product is a * c and whose sum is b. Factor by grouping terms in pairs.

3Quadratic formula

Concept Explanation

The quadratic formula is a universal method to find the roots of any quadratic equation. It is derived by completing the square on the standard quadratic equation.

Mathematical Representation

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Study Guideline: Substitute the values of a, b, and c carefully. The ± sign indicates that there will be two roots (one using + and one using -).

4Nature of roots (Discriminant)

Concept Explanation

The nature of the roots of ax² + bx + c = 0 is determined by the discriminant D = b² - 4ac. It tells us whether the roots are real, equal, or imaginary.

Mathematical Representation

D = b^2 - 4ac \implies \begin{cases} D > 0 & \text{Distinct Real Roots} \\ D = 0 & \text{Equal Real Roots} \\ D < 0 & \text{No Real Roots (Imaginary)} \end{cases}

Study Guideline: Check the sign of D first: if it is negative, you do not need to calculate real roots; if it is a perfect square, the roots are rational.