2.1.18
Discriminant woh expression hai Δ = b 2 − 4 a c poori tarah se decide karta hai ki roots real hain, equal hain, ya complex — bina equation solve kiye .
Intuition Ek Number Hume Sab Kuch Kyun Bata Deta Hai?Quadratic formula socho: x = 2 a − b ± b 2 − 4 a c
Sab kuch square root ke andar wali cheez pe depend karta hai: b 2 − 4 a c
Agar yeh positive hai → real square root le sakte ho → do alag-alag real answers
Agar yeh zero hai → ± ± 0
Agar yeh negative hai → negative ka square root → complex numbers aa jaate hain
Discriminant ek gatekeeper hai: yeh control karta hai ki tum real numbers mein rahoge ya complex territory mein jaoge.
Quadratic equation a x 2 + b x + c = 0 a = 0 discriminant hai:
Δ = b 2 − 4 a c
Completing the square se shuru karo (sabse fundamental approach):
a x 2 + b x + c = 0
Step 1: a a ( x 2 + a b x ) + c = 0
Kyun? Isse x 2
Step 2: Constant ko right side pe le jaao
a ( x 2 + a b x ) = − c
Step 3: Left side pe completing the square karo
a [ ( x + 2 a b ) 2 − ( 2 a b ) 2 ] = − c
Kyun? ( x + p ) 2 = x 2 + 2 p x + p 2 ( 2 a b ) 2
Step 4: Expand aur rearrange karo
a ( x + 2 a b ) 2 − a ⋅ 4 a 2 b 2 = − c
a ( x + 2 a b ) 2 = − c + 4 a b 2
Step 5: Right side pe common denominator laao
a ( x + 2 a b ) 2 = 4 a − 4 a c + b 2
( x + 2 a b ) 2 = 4 a 2 b 2 − 4 a c
Yahan critical moment hai: x
x + 2 a b = ± 4 a 2 b 2 − 4 a c = ± 2∣ a ∣ b 2 − 4 a c
Term b 2 − 4 a c Yeh positive, zero, ya negative hai — isi se humare solution ki nature decide hoti hai.
Kyunki hum generally a > 0
Kyun? Positive number ka square root real hota hai. ±
Graphically: Parabola x-axis ko do points pe cross karta hai .
Kyun? 0 = 0 ± 0
Graphically: Parabola exactly ek point pe x-axis ko touch karta hai (vertex pe).
Kyun? − k = i k k > 0 a + bi
Graphically: Parabola x-axis ko intersect nahi karta (poora oopar ya neeche rehta hai).
Worked example Example 1: Two Real RootsProblem: x 2 − 5 x + 6 = 0 Solution:
Identify karo: a = 1 , b = − 5 , c = 6
Discriminant calculate karo:
Δ = b 2 − 4 a c = ( − 5 ) 2 − 4 ( 1 ) ( 6 ) = 25 − 24 = 1 Yeh step kyun? Hume Δ
Kyunki Δ = 1 > 0 Two distinct real roots
Unhe dhundho:
x = 2 5 ± 1 = 2 5 ± 1 x 1 = 3 , x 2 = 2
Factorization yahan kyun kaam karega? Kyunki roots integers hain, toh ( x − 3 ) ( x − 2 ) = 0 Verification: ( 3 ) 2 − 5 ( 3 ) + 6 = 9 − 15 + 6 = 0 ( 2 ) 2 − 5 ( 2 ) + 6 = 4 − 10 + 6 = 0
Worked example Example 3: Complex RootsProblem: 2 x 2 + 3 x + 5 = 0
Solution:
Coefficients: a = 2 , b = 3 , c = 5
Discriminant:
Δ = 3 2 − 4 ( 2 ) ( 5 ) = 9 − 40 = − 31
Answer "no solution" kyun nahi hai? Real numbers mein koi solution nahi hai, lekin complex numbers mein do solutions hain.
Kyunki Δ < 0 Two complex conjugate roots
Formula apply karo:
x = 4 − 3 ± − 31 = 4 − 3 ± i 31 Yeh conjugate pairs mein kyun aate hain? Coefficients real hain, isliye complex roots α + i β α − i β Roots: x = − 4 3 ± 4 31 i
Worked example Example 4: Determining Parameter RangeProblem: m m x 2 − 4 x + 1 = 0
Solution:
Real roots ke liye: Δ ≥ 0 ≥ > Δ = 0
Yahan: a = m , b = − 4 , c = 1 Δ = ( − 4 ) 2 − 4 ( m ) ( 1 ) = 16 − 4 m
Condition:
16 − 4 m ≥ 0 16 ≥ 4 m m ≤ 4
Critical check: m = 0 Answer: m ∈ ( − ∞ , 4 ] ∖ { 0 } m ∈ ( − ∞ , 0 ) ∪ ( 0 , 4 ]
Zero kyun exclude karein? Agar m = 0 − 4 x + 1 = 0
Common mistake Mistake 1: "Δ < 0
Galat reasoning: "Square root ke andar negative → undefined → koi answer nahi"
Yeh sahi kyun lagta hai: Elementary math mein hum kehte hain "negative ka square root nahi le sakte."
Fix: Equation mein solutions hain — woh bas complex numbers hain. Discriminant hume nature batata hai (real vs complex), existence nahi.
Sahi statement: "Δ < 0 real solution nahi, lekin do complex conjugate solutions exist karte hain."
Common mistake Mistake 2:4 a
Galat: Δ = b 2 − c Δ = b 2 − a c
Kyun hota hai: Formula complicated lagta hai; students "main parts" yaad kar lete hain.
Fix: Quadratic formula ki structure yaad rakho: discriminant exactly woh hai jo square root ke andar hai.
b 2 − 4 a c 4 a
Mnemonic: "B -squared M inus F our A -C " → B²M-F-AC → B²-4AC
Common mistake Mistake 3: Negativeb
Galat: x 2 − 6 x + 5 = 0 Δ = 6 2 − 4 ( 1 ) ( 5 ) x = 2 a − b ± Δ b = 6
Kyun hota hai: − 6 x b = − 6 b = 6
Fix: Coefficient b b = − 6 Δ = ( − 6 ) 2 − 4 ( 1 ) ( 5 ) = 36 − 20 = 16 ( − 6 ) 2 = 36 x = 2 a − b ± Δ − b = − ( − 6 ) = + 6 − 6
Common mistake Mistake 4: "Δ = 0
Galat reasoning: "Zero matlab kuch nahi, toh koi roots nahi."
Kyun hota hai: "One repeated root" aur "no roots" ko confuse karna.
Fix: Δ = 0 multiplicity 2 wala ek real root milta hai. Algebraically ise do baar count kiya jaata hai: ( x − r ) 2 = 0 r
Geometric check: Parabola x-axis ko touch karta hai — intersection point IS hai.
Mnemonic Discriminant Decision Tree"Positive Parabola Pierces, Zero Touches, Negative Never"
Δ > 0 : P ositive → P arabola x-axis ko P ierce karta hai (2 points)Δ = 0 : Z ero → parabola ke Z ero crossings (1 point pe touch karta hai)Δ < 0 : N egative → x-axis ko kabhi N ahi touch karta (complex roots)
Alternative numeric mnemonic:
Δ = b 2 − 4 a c Bee squared minus Four Aces " (jaise playing cards: deck mein 4 aces hote hain)
Recall Ek 12-saal ke bachche ko explain karoSocho tum ek ball throw kar rahe ho. Equation tumhe batati hai ki ball kisi bhi time pe kitni oopar hai.
Discriminant aise hai jaise poochh rahe ho: "Kya ball zameen se takraayegi?"
Agar discriminant positive hai (bada number): Ball oopar jaati hai, neeche aati hai, aur DO alag-alag times pe zameen se takraati hai. (Tum ise raaste mein oopar jaate hue YA neeche aate hue pakad sakte ho.)
Agar discriminant zero hai: Ball zameen ko bas barely ek moment ke liye touch karti hai — jaise woh ek chhoti si bounce ke peak pe zameen pe tiki ho. Woh exactly ek time pe zameen ko graze karti hai.
Agar discriminant negative hai: Ball zameen ko bilkul nahi touch karti! Woh poore time hawa mein udti rehti hai. Real life mein iska matlab hai ki parabola (ball ka path) poori tarah zameen ke oopar rehti hai. Math mein hum kehte hain iske "imaginary" ya "complex" answers hain — jaise poochhna "oopar phenkee gayi ball underground tunnel se kab takraayegi?" Yeh ek valid math question hai, lekin answer koi real time nahi hai jo tum apni ghadi pe point kar sako.
Jaadu ki baat: Sirf EK number (b 2 − 4 a c
Quadratic Formula — discriminant woh expression hai jo square root ke andar haiComplex Numbers — tab aate hain jab Δ < 0 Completing the Square — woh method jo reveal karta hai ki discriminant ki yeh form kyun haiParabola and Axis of Symmetry — roots aur vertex ka geometric meaningVieta's Formulas — roots aur coefficients ke beech relationship: x 1 + x 2 = − b / a x 1 x 2 = c / a Factoring Quadratics — jab Δ Conjugate Root Theorem — explain karta hai ki complex roots a ± bi Graphing Quadratic Functions — x-intercepts ki number Δ
#flashcards/maths
Quadratic equation a x 2 + b x + c = 0 Δ = b 2 − 4 a c
Agar discriminant Δ > 0 Two distinct real roots
Agar discriminant Δ = 0 One repeated real root (equal roots)
Agar discriminant Δ < 0 Two complex conjugate roots
Discriminant roots ki nature kyun determine karta hai? Yeh quadratic formula mein square root ke andar wala expression hai; iska sign decide karta hai ki square root real hoga ya imaginary.
x 2 + 6 x + k = 0 k k = 9 Δ = 36 − 4 k = 0
Δ = 0 Parabola x-axis ko exactly ek point pe touch karta hai (vertex pe)
Δ < 0 Parabola x-axis ko intersect nahi karta (poora oopar ya neeche rehta hai)
Agar kisi quadratic ke liye Δ = 49 Two distinct real roots, kyunki
Δ > 0 aur
49 = 7 se
± ke saath do alag values milti hain
Discriminant explicitly dikhate hue quadratic formula likho x = 2 a − b ± Δ jahan
Δ = b 2 − 4 a c 2 x 2 − 5 x + 3 = 0 Δ = ( − 5 ) 2 − 4 ( 2 ) ( 3 ) = 25 − 24 = 1
Agar kisi quadratic ke complex roots 3 + 2 i 3 − 2 i Negative (Δ < 0
Ek quadratic perfect square trinomial hone ke liye discriminant ki kya value honi chahiye? Zero (Δ = 0 ( x − r ) 2
x 2 + m x + 4 = 0 m m ≤ − 4 m ≥ 4 Δ = m 2 − 16 ≥ 0 m 2 ≥ 16
Parabola crosses x-axis twice
Parabola touches x-axis once
