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CHAPTER 04

Quadratic Equations

Overview

Quadratic equation wo hai jisme variable ka square aata hai — ax² + bx + c = 0 (a ≠ 0). Is chapter mein equations ko factorisation, completing the square aur quadratic formula se solve karenge, aur discriminant (D = b² − 4ac) se pehle hi bata payenge ki roots real hain ya nahi. Word problems (age, speed, area) mein equation khud banana sabse important skill hai.

Key Concepts

  • 01Standard form: ax² + bx + c = 0, jahan a, b, c real numbers hain aur a ≠ 0.
  • 02Roots wo values hain jo equation ko satisfy karti hain — ek quadratic ke exactly do roots hote hain.
  • 03Factorisation: middle term ko do hisson mein todo taki common factor nikal aaye.
  • 04Quadratic formula se koi bhi quadratic solve ho jaati hai — factorisation na ho tab bhi.
  • 05Discriminant D = b² − 4ac: D > 0 → do distinct real roots; D = 0 → do equal real roots; D < 0 → koi real root nahi.
  • 06Word problems mein: variable define karo, condition se equation banao, solve karo, aur answer ko context me check karo (negative age/negative speed reject!).

Important Formulas

Standard form: ax² + bx + c = 0 (a ≠ 0)
Discriminant: D = b² − 4ac
Quadratic formula: x = (−b ± √D) / 2a
Sum of roots: α + β = −b/a
Product of roots: αβ = c/a

Practice Questions

  1. 01x² − 5x + 6 = 0 ko factorisation se solve karo.
  2. 022x² − 4x + 3 = 0 ke roots ka nature discriminant se batao.
  3. 03Wo do consecutive positive integers nikalo jinke square ka sum 365 hai.
  4. 04Ek train samay se 1 ghanta late chalti hai; 40 km/h tez chal kar wo 240 km phir se time par pahunchti hai — normal speed nikalo.
  5. 05Ek quadratic ke roots ka sum 2 aur product −8 hai — equation banao.
Summary

Quadratic ax² + bx + c = 0 ko teen tarike se solve karte hain: factorisation, completing the square, aur formula x = (−b ± √D)/2a. Discriminant D = b² − 4ac roots ka nature batata hai — positive: do alag real roots, zero: equal roots, negative: no real roots. Sum = −b/a, product = c/a — isse equation bhi bana sakte hain: x² − (sum)x + product = 0.

Handwritten notes
Handwritten — class notes
Handwritten — class notes
Diagrams + workings
Diagrams + workings

Chapter-end quiz

5 MCQs

Option tap karo — turant pata chalega sahi ya galat, explanation ke saath.

  1. Q1.2x² − 4x + 3 = 0 ka discriminant kitna hai?

  2. Q2.x² − 9 = 0 ke roots kya hain?

  3. Q3.Agar D = 0 ho to roots kaisi hoti hain?

  4. Q4.x² − 5x + 6 = 0 ke roots ka sum kitna hai?

  5. Q5.Jinke roots 2 aur −3 hain, wo equation kaunsi hai?

Is chapter ka doubt?
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