Quadratic Equations – Nature of Roots
Quadratic Equations JEE Main Mathematics ka ek fundamental topic hai. Is topic mein sirf roots find karna hi important nahi hai, balki bina roots calculate kiye roots ki nature determine karna bhi bahut important hai.
JEE Main mein questions aksar discriminant, equal roots, distinct real roots, no real roots, roots ke signs aur parameter-based conditions par based hote hain.
Is chapter ka sabse important tool hai:
Discriminant: D = b² − 4ac
Agar discriminant ka concept clear hai, to quadratic equation ke bahut saare questions seconds mein solve kiye ja sakte hain.
1. Basic Form of Quadratic Equation
Ek quadratic equation ka standard form hota hai:
ax² + bx + c = 0
Yahan:
- a, b, c constants hain.
- a ≠ 0 hona compulsory hai.
- Equation ke roots ko generally α aur β se represent kiya jaata hai.
Example:
x² − 5x + 6 = 0
Is equation mein:
a = 1, b = −5, c = 6
Quadratic equation ke roots ko solve karne ke liye quadratic formula use kiya ja sakta hai, lekin nature of roots ke liye humein directly discriminant use karna hota hai.
2. Discriminant ⭐
Quadratic equation:
ax² + bx + c = 0
ka discriminant hota hai:
D = b² − 4ac
D roots ki nature decide karta hai.
🔥 Golden Rule
| Discriminant | Nature of Roots |
|---|---|
| D > 0 | 2 distinct real roots |
| D = 0 | 2 equal real roots |
| D < 0 | No real roots |
Is table ko JEE Main ke liye perfectly memorize karna important hai.
3. Case 1: D > 0
Agar:
b² − 4ac > 0
to quadratic equation ke roots:
Real + Unequal
Yaani equation ke 2 distinct real roots honge.
Example
x² − 5x + 6 = 0
Yahan:
a = 1, b = −5, c = 6
Discriminant:
D = (−5)² − 4(1)(6)
D = 25 − 24 = 1
Since:
D = 1 > 0
Therefore roots are real and unequal.
Actually, equation factorize karne par:
(x − 2)(x − 3) = 0
So roots are:
α = 2, β = 3
4. Case 2: D = 0
Agar:
b² − 4ac = 0
to roots real and equal hote hain.
Is case mein:
α = β
Equal Root Formula
Quadratic formula:
x = (−b ± √D)/2a
Jab D = 0:
√D = 0
Therefore:
α = β = −b/2a
Example
x² − 6x + 9 = 0
Discriminant:
D = (−6)² − 4(1)(9)
D = 36 − 36 = 0
Therefore roots are equal.
Equal root:
x = −(−6)/(2 × 1) = 3
Hence:
α = β = 3
5. Case 3: D < 0
Agar:
b² − 4ac < 0
to quadratic equation ke koi real roots nahi hote.
Roots complex ho sakte hain, lekin real number system mein roots exist nahi karte.
Example
x² + 2x + 5 = 0
Yahan:
a = 1, b = 2, c = 5
Discriminant:
D = 2² − 4(1)(5)
D = 4 − 20 = −16
Since:
D < 0
Therefore equation ke no real roots hain.
6. Nature of Roots – Parabola Se Samjho
Quadratic equation ka graph ek parabola hota hai. Roots woh x-values hoti hain jahan parabola x-axis ko intersect karta hai.
D > 0
Parabola x-axis ko 2 different points par cut karta hai.
2 real unequal roots
Visual memory:
∪
/ \
------●-------●------ x-axis
D = 0
Parabola x-axis ko exactly one point par touch karta hai. Ye point double root represent karta hai.
2 equal real roots
∪
/ \
--------●---------- x-axis
D < 0
Parabola x-axis ko meet nahi karta.
No real roots
∪
/ \
/ \
---------------- x-axis
🔥 Visual Shortcut
Cut → D > 0
Touch → D = 0
No Meet → D < 0
7. Formula of Roots
Quadratic equation:
ax² + bx + c = 0
ke roots:
x = (−b ± √D)/2a
where:
D = b² − 4ac
Ab discriminant ke three cases ko formula se directly understand kar sakte hain:
- D > 0: √D positive real number hai, so ± se two different real values milti hain.
- D = 0: √D = 0, so dono roots same value dete hain.
- D < 0: √D real nahi hai, so roots real nahi honge.
8. Shortcut for Equal Roots ⭐
Agar question mein directly bola ho ki quadratic equation ke roots equal hain, to roots calculate karne ki zarurat nahi hai.
Direct condition:
D = 0
Therefore:
b² − 4ac = 0
Ya:
b² = 4ac
Ye condition parameter-based questions mein extremely useful hai.
9. Sum and Product of Roots
Agar roots α and β hain for:
ax² + bx + c = 0
then:
α + β = −b/a
αβ = c/a
Ye formulas roots ke signs determine karne mein bahut useful hain.
10. Conditions for Roots to Have Same Sign
Agar roots ka same sign hai, to unka product positive hona chahiye:
αβ > 0
Since:
αβ = c/a
therefore:
c/a > 0
Lekin sirf product positive hona enough nahi hai to decide whether both roots positive hain ya both negative. Sum ka sign bhi check karna hoga.
Both Roots Positive
Agar:
α > 0 and β > 0
then:
α + β > 0
and:
αβ > 0
Therefore:
−b/a > 0
c/a > 0
Both Roots Negative
Agar:
α < 0 and β < 0
then:
α + β < 0
and:
αβ > 0
Therefore:
−b/a < 0
c/a > 0
11. Roots Have Opposite Signs
Agar ek root positive aur doosra negative hai, to product negative hoga:
αβ < 0
Since:
αβ = c/a
therefore:
c/a < 0
⭐ JEE Shortcut
c/a < 0 → Roots have opposite signs
Example:
2x² + 3x − 5 = 0
Here:
c/a = −5/2 < 0
Therefore one root positive aur one root negative hoga.
12. Parameter-Based Questions 🔥
JEE Main mein quadratic equation mein unknown parameter diya hota hai, aur question roots ki nature ke basis par parameter ki value ya range pooch sakta hai.
Example
For:
x² − 2(k + 1)x + k² = 0
find the value of k for equal roots.
Step 1: Equal Roots Condition
D = 0
Step 2: Discriminant Calculate Karo
Here:
a = 1
b = −2(k + 1)
c = k²
Therefore:
[−2(k + 1)]² − 4(1)(k²) = 0
4(k + 1)² − 4k² = 0
Divide by 4:
(k + 1)² − k² = 0
Expand:
k² + 2k + 1 − k² = 0
2k + 1 = 0
Therefore:
k = −1/2
13. Real Roots Ke Liye Condition
Agar question kahe ki quadratic equation ke real roots hone chahiye, to discriminant:
D ≥ 0
hona chahiye.
Example
Suppose:
x² + 2x + k = 0
has real roots.
Condition:
D ≥ 0
2² − 4(1)(k) ≥ 0
4 − 4k ≥ 0
1 − k ≥ 0
Hence:
k ≤ 1
🔥 Important
Real roots → D ≥ 0
Distinct real roots → D > 0
Equal roots → D = 0
14. Solved JEE Main-Level Question
Question
For what values of k does:
x² − 4x + k = 0
have two distinct real roots?
Step 1: Identify Condition
Two distinct real roots ke liye:
D > 0
Step 2: Calculate D
Here:
a = 1, b = −4, c = k
Therefore:
D = (−4)² − 4(1)(k)
D = 16 − 4k
Step 3: Apply D > 0
16 − 4k > 0
Divide by 4:
4 − k > 0
Therefore:
k < 4
✅ Answer: k < 4
15. Common JEE Traps ⚠️
| Wrong Thinking | Correct Concept |
|---|---|
| D ≥ 0 means distinct roots | D > 0 → distinct real roots |
| D = 0 means no roots | D = 0 → equal real roots |
| D < 0 means equal roots | D < 0 → no real roots |
| Only b² matters | Always calculate b² − 4ac |
| c/a > 0 alone proves both roots positive | Sum and product both need to be checked |
| c/a < 0 means both roots are negative | c/a < 0 means opposite signs |
16. JEE Main Quick-Solving Strategy
Nature of roots ke questions mein unnecessary quadratic formula use karne ki zarurat nahi hoti. Pehle question mein given condition identify karo.
- Equal roots? → D = 0
- Two distinct real roots? → D > 0
- Real roots? → D ≥ 0
- No real roots? → D < 0
- Opposite signs? → c/a < 0
- Both positive? → α + β > 0 and αβ > 0
- Both negative? → α + β < 0 and αβ > 0
Is approach se parameter-based questions bahut quickly solve ho jaate hain.
17. Important Formula Sheet
| Concept | Formula / Condition |
|---|---|
| Quadratic Equation | ax² + bx + c = 0 |
| Discriminant | D = b² − 4ac |
| Roots | x = (−b ± √D)/2a |
| Real Roots | D ≥ 0 |
| Distinct Real Roots | D > 0 |
| Equal Real Roots | D = 0 |
| No Real Roots | D < 0 |
| Sum of Roots | α + β = −b/a |
| Product of Roots | αβ = c/a |
| Opposite Signs | c/a < 0 |
18. One-Minute Revision
Agar exam se just pehle is topic ko revise karna ho, to sirf ye points yaad rakho:
D = b² − 4ac
D > 0 → Real + Unequal
D = 0 → Real + Equal
D < 0 → No Real Roots
Roots = (−b ± √D)/2a
α + β = −b/a
αβ = c/a
c/a < 0 → Opposite Signs
Real Roots → D ≥ 0
Distinct Real Roots → D > 0
Equal Roots → D = 0
19. 🔥 Final Revision Box
Quadratic Equation → ax² + bx + c = 0
a ≠ 0
D = b² − 4ac
D > 0 → Real + Unequal
D = 0 → Real + Equal
D < 0 → No Real Roots
Roots → (−b ± √D)/2a
α + β = −b/a
αβ = c/a
c/a < 0 → Roots have opposite signs
Real Roots → D ≥ 0
Distinct Real Roots → D > 0
Equal Roots → D = 0
20. Practice Questions
- Determine the nature of roots of x² − 7x + 12 = 0.
- Find the value of k for which x² + kx + 9 = 0 has equal roots.
- Find the range of k for which x² − 2x + k = 0 has real roots.
- Determine whether the roots of 3x² + 5x − 2 = 0 have the same or opposite signs.
- For what values of k will x² − 6x + k = 0 have two distinct real roots?
- Find the nature of roots of 2x² + 4x + 5 = 0.
- If both roots of ax² + bx + c = 0 are positive, what conditions should be satisfied by their sum and product?
- If c/a < 0, what can you conclude about the signs of the roots?
21. PDF Notes
Neeche Quadratic Equations – Nature of Roots ke revision notes/PDF ko embed kiya ja sakta hai.
22. Frequently Asked Questions (FAQs)
Q1. Discriminant kya hota hai?
Quadratic equation ax² + bx + c = 0 ke liye discriminant D = b² − 4ac hota hai. Ye roots ki nature determine karta hai.
Q2. D = 0 hone par roots kaise hote hain?
D = 0 hone par roots real and equal hote hain. Equal root −b/2a hota hai.
Q3. D > 0 ka kya meaning hai?
D > 0 ka matlab hai equation ke two distinct real roots hain.
Q4. D < 0 hone par kya hota hai?
D < 0 hone par quadratic equation ke no real roots hote hain.
Q5. Real roots ke liye discriminant ki condition kya hai?
Real roots ke liye: D ≥ 0
Q6. Opposite signs ke roots kaise identify karein?
Agar: αβ = c/a < 0, to roots opposite signs ke honge — ek positive aur doosra negative.
Q7. Kya c/a > 0 ka matlab dono roots positive hain?
Nahi. c/a > 0 sirf batata hai ki roots ka product positive hai, isliye roots same sign ke hain. Dono positive ya dono negative decide karne ke liye α + β = −b/a ka sign bhi check karna hota hai.
Q8. Parabola aur roots ka kya relation hai?
Quadratic equation ka graph parabola hota hai. Agar parabola x-axis ko two points par cut karta hai to D > 0, ek point par touch karta hai to D = 0, aur x-axis ko meet nahi karta to D < 0.
Final Thoughts
Nature of Roots ko master karne ke liye sabse important concept discriminant hai. Har baar roots solve karne ke bajay pehle question ki condition identify karo aur directly discriminant apply karo.
JEE Main ke liye teen golden conditions hamesha yaad rakho:
D > 0 → Real + Unequal
D = 0 → Real + Equal
D < 0 → No Real Roots
Iske saath α + β = −b/a aur αβ = c/a ko combine karna seekh lo. Phir roots ke signs, parameter-based questions aur many JEE Main problems ko bina roots calculate kiye solve kiya ja sakta hai.
Discriminant samjho → Condition identify karo → Inequality solve karo → Answer verify karo. Ye simple approach quadratic equations ke questions mein speed aur accuracy dono improve karegi.