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Quadratic Equations and Linear Inequalities

NIOS · Class 12 · Mathematics

Quick revision notes for Quadratic Equations and Linear Inequalities — NIOS Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

45 questions25 flashcards5 concepts

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Key Topics to Revise

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1. Quadratic Equations — Basics and Roots

  • A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. Here 'a' is the leading coefficient, 'b' is the middle coefficient, and 'c' is the constant term.
  • A root (or solution) is a value of x that satisfies the equation when substituted into it.
  • Every quadratic equation has EXACTLY 2 roots (counting multiplicity) — this follows from the Fundamental Theorem of Algebra.
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2. Solving Quadratic Equations by Factorization

  • Factorization method involves splitting the middle term (bx) into two parts whose product equals a × c and whose sum equals b.
  • After splitting, group the terms, take common factors, and express as a product of two linear factors.
  • Set each factor equal to zero and solve — this gives the two roots.
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3. Solving Quadratic Equations by Quadratic Formula (Sridharacharya's Formula)

  • The quadratic formula gives the roots of ax² + bx + c = 0 directly using the coefficients a, b, c.
  • The discriminant D = b² - 4ac determines the NATURE of roots BEFORE actually computing them.
  • If D > 0: two distinct real roots. If D = 0: two equal real roots (each equal to -b/2a). If D < 0: two complex conjugate roots.
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4. Relation Between Roots and Coefficients

  • For the quadratic equation ax² + bx + c = 0 with roots α and β: Sum of roots = α + β = -b/a, Product of roots = αβ = c/a.
  • These relations allow us to find expressions involving α and β WITHOUT actually solving for α and β separately.
  • To form a new quadratic equation with given roots p and q: x² - (p + q)x + pq = 0.

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Full Notes

Key Concepts

A polynomial equation of the secondA valueA technique to solve quadratic equationsA universal formula to find rootsThe expression D = b²

Frequently Asked Questions

What are the important topics in Quadratic Equations and Linear Inequalities for NIOS Class 12 Mathematics?
Key topics in Quadratic Equations and Linear Inequalities include Mind map showing the structure, components, and key characteristics of a quadratic equation, Step-by-step flowchart showing the factorization method for solving quadratic equations, Flowchart showing the quadratic formula method and how the discriminant determines root nature. These are the concepts NIOS Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Quadratic Equations and Linear Inequalities — NIOS Class 12 Mathematics?
Understand the core concepts first, then work through the 45 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

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