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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.

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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?
Quadratic Equations and Linear Inequalities covers several key topics that are frequently asked in NIOS Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Quadratic Equations and Linear Inequalities — NIOS Class 12 Mathematics?
Start by understanding all key concepts. Practise previous year questions from this chapter. Revise formulas and definitions regularly. Use flashcards for quick revision before the exam.

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