Quadratic equations class 10 ke Maths ka Chapter 4 hai aur board exam mein isse 4-5 marks pakke aate hain. Is chapter ka poora khel teen cheezon par hai: samikaran ko pehchanna, use hal karna (roots nikaalna) aur discriminant se roots ki prakriti bataana. Aaiye step-by-step samajhte hain.
Quadratic Equations Class 10 — Dvighaat Samikaran Kya Hai?
Jis samikaran mein variable ki sabse badi ghaat (power) 2 ho, use quadratic (dvighaat) samikaran kehte hain.
Standard form: ax² + bx + c = 0, jahan a ≠ 0
- a, b, c coefficients hain (sankhyaen)
- a ≠ 0 hona zaroori hai — varna yeh linear samikaran ban jaayega
Udaharan: 2x² + 5x + 3 = 0 ek quadratic equation hai (a=2, b=5, c=3). Lekin 4x + 7 = 0 quadratic NAHI hai (x² ka term nahi hai).
Samikaran ke hal (x ke maan) ko roots (mool) kehte hain — quadratic equation ke adhiktam 2 roots hote hain.
Factorization Se Hal Karna
Quadratic equations class 10 mein sabse aasaan tareeka — jab samikaran factor ho jaaye:
Udaharan 1: x² + 5x + 6 = 0
- 6 ke aise do gunankhand dhoondo jinka yog 5 ho: 2 aur 3
- x² + 2x + 3x + 6 = 0
- x(x + 2) + 3(x + 2) = 0
- (x + 2)(x + 3) = 0
- Roots: x = -2, x = -3
Trick: constant term (c) ke factors dhoondo, jinka yog b ke barabar ho.
Poorna Varg Banakar Hal (Completing the Square)
Jab factorization mushkil lage:
Udaharan 2: x² + 4x + 1 = 0
- x² + 4x = -1
- Dono taraf (4/2)² = 4 jodo: x² + 4x + 4 = -1 + 4
- (x + 2)² = 3
- x + 2 = ±√3
- Roots: x = -2 + √3, x = -2 – √3
Quadratic Equations Class 10 — Dvighaat Sutra (Quadratic Formula)
Har quadratic equation ko hal karne ka pakka formula:
x = [-b ± √(b² – 4ac)] / 2a
Udaharan 3: 2x² – 7x + 3 = 0 (a=2, b=-7, c=3)
- x = [7 ± √(49 – 24)] / 4
- x = [7 ± √25] / 4 = [7 ± 5] / 4
- Roots: x = 3, x = 1/2
Formula mein b ka chinh (sign) dhyaan se rakhein — yahi sabse common mistake hai.
Vivechak (Discriminant) — D = b² – 4ac
Bina hal kiye roots ki prakriti bataane ka tareeka:
- D > 0: do alag-alag real roots (jaise Udaharan 3 mein D=25)
- D = 0: do barabar real roots (ek hi root do baar)
- D < 0: koi real root nahi (imaginary roots)
Udaharan 4: x² + 4x + 4 = 0 mein D = 16 – 16 = 0 → do barabar roots, x = -2.
Udaharan 5: x² + x + 1 = 0 mein D = 1 – 4 = -3 < 0 → koi real root nahi.
Board tip: “Roots ki prakriti gyaat kijiye” — iska matlab sirf D nikaalkar upar waali teen sthitiyon mein se ek likhna hai.
Moolon Ka Yog Aur Gunfal
Bina roots nikaale bhi:
- Roots ka yog (sum) = -b/a
- Roots ka gunfal (product) = c/a
Udaharan 6: 3x² – 5x + 2 = 0 mein roots ka yog = 5/3 aur gunfal = 2/3.
Exam Ke Liye Key Points
- Standard form ax² + bx + c = 0 (a ≠ 0) — pehchanna seekhein.
- Quadratic formula: x = [-b ± √(b²-4ac)]/2a — chinh ki galti na karein.
- D > 0 / D = 0 / D < 0 — teeno sthitiyan example ke saath yaad rakhein.
- Word problems mein pehle samikaran banaana seekhein (jaise: “do sankhyaon ka yog 12, gunfal 35” → x² – 12x + 35 = 0).
Aur study notes ke liye https://jagatindia.com/category/students/ par jaayein.