Quadratic Equations Class 11: Important Formulas, Roots & JEE Main Shortcuts

Quadratic Equations – JEE Main Ke Most Important Concepts

Quadratic Equations Class 11 Mathematics ka ek important topic hai. Is chapter mein aap quadratic formula, discriminant, nature of roots, roots aur coefficients ke relations, sign of roots aur equation formation jaise concepts seekhte ho.

JEE Main mein in concepts ka use direct questions ke saath parameter-based questions aur algebraic expressions simplify karne mein bhi hota hai. Agar aap roots ko directly calculate karne ke bajaye unke sum aur product ka use karna seekh jaate ho, toh kai questions kam steps mein solve ho sakte hain.

Chalo, important concepts ko examples aur exam-oriented shortcuts ke saath samajhte hain.

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1. Quadratic Equation – Basic Formula

Standard Form

Quadratic equation ka standard form hota hai:

ax² + bx + c = 0, where a ≠ 0

Yahan:

• a: x² ka coefficient.
• b: x ka coefficient.
• c: Constant term.

Equation quadratic tab hoti hai jab x ki highest power 2 ho aur x² ka coefficient zero na ho.

Example

Consider the equation:

3x² − 5x + 2 = 0

Is equation mein:

• a = 3
• b = −5
• c = 2

Sign ka dhyan rakho. Coefficient b = −5 hai, 5 nahi.

Quadratic Formula

Quadratic equation ax² + bx + c = 0 ke roots find karne ka formula hai:

x = (−b ± √(b² − 4ac)) / 2a

Roots ko generally α aur β se represent kiya jaata hai.

Is formula mein b² − 4ac ko discriminant kehte hain, jise D se denote karte hain.

Isliye quadratic formula ko is tarah bhi likh sakte hain:

x = (−b ± √D) / 2a

Example: Find the Roots

Solve:

x² − 5x + 6 = 0

Here, a = 1, b = −5 and c = 6.

Using the quadratic formula:

x = [−(−5) ± √((-5)² − 4(1)(6))] / 2

x = [5 ± √(25 − 24)] / 2

x = (5 ± 1) / 2

Therefore:

• x = (5 + 1)/2 = 3
• x = (5 − 1)/2 = 2

Roots are 2 and 3.

JEE Tip: Quadratic formula use karte waqt b ka sign carefully substitute karo. Agar b negative hai, toh −b positive ho jaayega.

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2. Discriminant – Nature of Roots

Discriminant quadratic equation ka ek important expression hai jo batata hai ki equation ke roots real hain, equal hain ya complex.

D = b² − 4ac

Discriminant calculate karne ke liye a, b aur c ki values standard form se identify karo.

Nature of Roots Table

Discriminant	Nature of Roots
D > 0	Two distinct real roots
D = 0	Two equal real roots
D < 0	Two non-real complex conjugate roots, for real coefficients

Important Conditions for JEE Main

• Real and distinct roots: D > 0.
• Equal real roots: D = 0.
• Real roots: D ≥ 0.

Agar question mein sirf real roots ki condition di hai, toh D ≥ 0 use karo. Agar distinct real roots maange hain, toh D > 0 use karna hai.

Example 1: Find the Nature of Roots

Determine the nature of roots of:

2x² − 5x + 2 = 0

Here, a = 2, b = −5 and c = 2.

D = b² − 4ac

D = (−5)² − 4(2)(2)

D = 25 − 16 = 9

Since D > 0, the equation has two distinct real roots.

Example 2: Real Roots Ke Liye Condition

Find the values of k for which the equation has real roots:

x² − 4x + k = 0

For real roots:

D ≥ 0

(−4)² − 4(1)(k) ≥ 0

16 − 4k ≥ 0

−4k ≥ −16

Dividing by −4 reverses the inequality:

k ≤ 4

Therefore, equation ke roots real honge jab k ≤ 4.

Rational Roots Ke Liye Important Note

Agar quadratic equation ke coefficients integers hain, toh rational roots ke liye discriminant ka non-negative perfect square hona necessary hai. Agar D perfect square nahi hai, toh roots rational nahi honge.

For example, x² − 5x + 6 = 0 ka discriminant 1 hai, jo perfect square hai, aur roots rational hain.

Note: D ka perfect square hona rational roots ki condition hai, jab equation integer coefficients mein ho. Real roots ke liye sirf D ≥ 0 sufficient hai.

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3. Relation Between Roots and Coefficients

Ye quadratic equations ka sabse useful concept hai. Agar equation ke roots α aur β hain, toh unke sum aur product ko coefficients se directly calculate kar sakte hain.

Consider:

ax² + bx + c = 0

Let its roots be α and β.

Sum of Roots

α + β = −b/a

Product of Roots

αβ = c/a

In relations ko Vieta's formulas bhi kaha jaata hai.

Example

For the equation:

2x² − 7x + 3 = 0

Here, a = 2, b = −7 and c = 3.

Sum of roots:

α + β = −(−7)/2 = 7/2

Product of roots:

αβ = 3/2 = 3/2

Notice that roots ko individually find kiye bina sum aur product mil gaye.

Useful Identity 1: α² + β²

We know:

(α + β)² = α² + 2αβ + β²

Rearranging:

α² + β² = (α + β)² − 2αβ

Ye identity tab useful hoti hai jab question mein roots ke squares ka sum poocha jaaye.

Useful Identity 2: (α − β)²

We know:

(α − β)² = α² − 2αβ + β²

Using α² + β² = (α + β)² − 2αβ:

(α − β)² = (α + β)² − 4αβ

Iska use roots ke difference ke square ko find karne mein hota hai.

Useful Identity 3: Reciprocal Sum

Agar αβ ≠ 0, toh:

1/α + 1/β = (β + α)/(αβ)

Therefore:

1/α + 1/β = (α + β)/αβ

Ye identity reciprocal roots ke sum ko quickly calculate karne mein help karti hai.

JEE Shortcut

Agar question mein α + β, αβ, α² + β², α − β ya reciprocal roots ka sum diya ho ya poocha gaya ho, toh pehle Vieta's formulas check karo.

Har baar quadratic formula se roots nikalna zaroori nahi hota.

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4. Sign of Roots – Important Conditions

Quadratic equation ke roots positive hain, negative hain ya opposite signs ke hain, ye questions mein directly poocha ja sakta hai. Iske liye sum aur product ka use karte hain.

Let the real roots be α and β.

Condition	Conclusion
αβ < 0	Roots have opposite signs
αβ > 0 and α + β > 0	Both roots are positive
αβ > 0 and α + β < 0	Both roots are negative
αβ = 0	At least one root is zero

Important: In conditions ko use karne se pehle confirm karo ki roots real hain.

Case 1: Opposite-Sign Roots

Agar αβ < 0, toh ek root positive aur doosra negative hoga.

Example:

x² − x − 6 = 0

Sum of roots = 1.

Product of roots = −6.

Since product negative hai, roots ke signs opposite honge. Actually, equation factorize karne par:

(x − 3)(x + 2) = 0

Roots 3 aur −2 hain.

Case 2: Both Roots Positive

Agar:

• αβ > 0
• α + β > 0

Toh dono real roots positive honge.

Example:

x² − 5x + 6 = 0

Sum = 5 and product = 6.

Dono roots positive hain: 2 and 3.

Case 3: Both Roots Negative

Agar:

• αβ > 0
• α + β < 0

Toh dono real roots negative honge.

Example:

x² + 5x + 6 = 0

Sum = −5 and product = 6.

Roots −2 and −3 hain.

JEE Main Trap

Sirf αβ > 0 dekhkar ye conclude nahi kar sakte ki roots positive hain. Product positive hone ka matlab hai dono roots same sign ke hain, assuming they are real. Dono positive ya dono negative hain, ye sum se decide hota hai.

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5. Formation of Quadratic Equation

Agar roots α aur β diye gaye hain, toh quadratic equation ko sum aur product ki help se form kar sakte hain.

Monic quadratic equation ka formula hai:

x² − (α + β)x + αβ = 0

Yahan monic ka matlab hai x² ka coefficient 1 hai.

Example: Roots Are 2 and 5

Given roots:

α = 2, β = 5

Sum of roots:

α + β = 2 + 5 = 7

Product of roots:

αβ = 2 × 5 = 10

Equation:

x² − (sum)x + product = 0

x² − 7x + 10 = 0

Ye required quadratic equation hai.

Verification

x² − 7x + 10 = 0

(x − 2)(x − 5) = 0

Therefore, x = 2 or x = 5.

Given roots correctly reproduce ho gaye.

What If the Leading Coefficient Is Not 1?

Equation ko kisi bhi non-zero constant se multiply kar sakte hain bina roots badle.

For example:

x² − 7x + 10 = 0

Is equation ko 2 se multiply karne par:

2x² − 14x + 20 = 0

Is equation ke roots bhi 2 aur 5 hi hain.

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6. Equal Roots and Parameter-Based Questions

Jab question mein equal roots ki condition di ho aur kisi parameter ki value find karni ho, tab discriminant sabse useful method hota hai.

Equal roots ke liye:

D = b² − 4ac = 0

Example: Find k for Equal Roots

Find k if the equation:

x² − 6x + k = 0

has equal roots.

Here:

• a = 1
• b = −6
• c = k

For equal roots:

D = 0

(−6)² − 4(1)(k) = 0

36 − 4k = 0

4k = 36

k = 9

Verify the Result

Substitute k = 9:

x² − 6x + 9 = 0

(x − 3)² = 0

Therefore, both roots are 3.

Equal roots ki condition satisfy ho rahi hai.

Parameter-Based JEE Shortcut

Question mein agar phrases diye hon, jaise:

• Equal roots
• Repeated roots
• Roots coincide
• Find the value of parameter k for equal roots

Toh sabse pehle discriminant ko zero set karne ka option check karo.

Agar real roots maange gaye hain, toh D ≥ 0 use karo. Agar two distinct real roots maange gaye hain, toh D > 0 use karo.

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7. JEE Main-Level Solved Question

Question: Equation 2x² − 7x + 3 = 0 ke roots α aur β hain. Find α² + β².

Options:

1. 25/4
2. 37/4
3. 49/4
4. 13/2

Step 1: Identify the Coefficients

Given equation:

2x² − 7x + 3 = 0

Therefore:

• a = 2
• b = −7
• c = 3

Step 2: Find the Sum of Roots

Using:

α + β = −b/a

α + β = −(−7)/2

α + β = 7/2

Step 3: Find the Product of Roots

Using:

αβ = c/a

αβ = 3/2

Step 4: Apply the Identity

We need α² + β².

Use:

α² + β² = (α + β)² − 2αβ

Substitute the values:

α² + β² = (7/2)² − 2(3/2)

= 49/4 − 3

= 49/4 − 12/4

α² + β² = 37/4

Correct Answer: Option B) 37/4

Why Is This the Best Method?

Is question mein roots ko individually calculate karna unnecessary hai. Sum aur product of roots use karne se answer directly mil jaata hai. JEE Main mein aise shortcuts calculation time reduce kar sakte hain.

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8. Common Mistakes in Quadratic Equations

• Wrong sign of b: Formula mein −b hota hai. Agar b = −7 hai, toh −b = +7 hoga.
• Discriminant conditions confuse karna: Real roots ke liye D ≥ 0; distinct real roots ke liye D > 0; equal roots ke liye D = 0.
• Sum of roots ka sign galat lena: α + β = −b/a hota hai, b/a nahi.
• Product of roots ka formula bhoolna: αβ = c/a.
• Roots ke sign ka incorrect conclusion: Positive product ka matlab same sign; sum se sign decide hota hai.
• Equation formation mein sign mistake: Roots α aur β ke liye x² − (α + β)x + αβ = 0.
• Zero product ko ignore karna: αβ = 0 ka matlab at least one root zero hai.
• Real-root assumption bhoolna: Sum aur product ki sign conditions se roots ke signs conclude karne ke liye roots real hone chahiye.

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9. Quadratic Equations Formula Sheet – Quick Revision

Standard Form and Formula

• Standard form: ax² + bx + c = 0, where a ≠ 0.
• Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.
• Discriminant: D = b² − 4ac.

Nature of Roots

• D > 0 → Two distinct real roots.
• D = 0 → Two equal real roots.
• D < 0 → Two non-real complex conjugate roots for real coefficients.
• D ≥ 0 → Real roots.

Relations Between Roots and Coefficients

• Sum: α + β = −b/a.
• Product: αβ = c/a.
• α² + β² = (α + β)² − 2αβ.
• (α − β)² = (α + β)² − 4αβ.
• 1/α + 1/β = (α + β)/αβ, provided αβ ≠ 0.

Sign of Roots

• αβ < 0 → Opposite signs, for real roots.
• αβ > 0 and α + β > 0 → Both positive, for real roots.
• αβ > 0 and α + β < 0 → Both negative, for real roots.
• αβ = 0 → At least one root is zero.

Equation Formation

If roots are α and β:

x² − (α + β)x + αβ = 0

Equal Roots

b² − 4ac = 0

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10. One-Minute Revision

1. Standard form: ax² + bx + c = 0, where a ≠ 0.
2. Quadratic formula: x = (−b ± √D)/2a.
3. Discriminant: D = b² − 4ac.
4. Real roots: D ≥ 0.
5. Equal roots: D = 0.
6. Distinct real roots: D > 0.
7. Sum of roots: α + β = −b/a.
8. Product of roots: αβ = c/a.
9. α² + β² = (α + β)² − 2αβ.
10. Equation from roots: x² − (α + β)x + αβ = 0.
11. Opposite-sign real roots: αβ < 0.
12. Equal roots ke parameter questions mein D = 0 use karo.

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11. Practice Questions for JEE Main

In questions ko pehle khud solve karo. Uske baad answer check karna.

Question 1

Find the discriminant of x² − 6x + 5 = 0.

A) 4

B) 16

C) 36

D) 56

Show Answer

D = b² − 4ac = (−6)² − 4(1)(5) = 36 − 20 = 16 (Option B).

Question 2

If α and β are roots of 3x² − 8x + 2 = 0, find α + β.

Show Answer

α + β = −b/a = −(−8)/3. Therefore, α + β = 8/3.

Question 3

If α and β are roots of x² − 4x + 1 = 0, find α² + β².

Show Answer

Sum = 4 and product = 1. Therefore, α² + β² = (α + β)² − 2αβ = 16 − 2 = 14.

Question 4

Find k if x² + 4x + k = 0 has equal roots.

Show Answer

For equal roots, D = 0. Thus, 4² − 4(1)(k) = 0, so 16 − 4k = 0. Therefore, k = 4.

Question 5

Form a quadratic equation whose roots are −2 and 7.

Show Answer

Sum = −2 + 7 = 5 and product = −14. Equation: x² − 5x − 14 = 0. Answer: x² − 5x − 14 = 0.

Question 6

For a quadratic equation with real roots α and β, if αβ < 0, what can you conclude about their signs?

Show Answer

The product is negative, so one root is positive and the other is negative. The roots have opposite signs.

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12. Download or View Your Quadratic Equations Notes PDF

Agar aapke paas Quadratic Equations ki PDF notes file hai, toh uska direct URL neeche diye gaye placeholder ki jagah paste kar sakte ho. Isse students article ke andar hi notes view kar paayenge.

Note: PDF embed kaam karne ke liye YOUR_PDF_LINK ko actual publicly accessible PDF URL se replace karna hoga.

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13. Frequently Asked Questions (FAQs)

Q1. Quadratic equation ka standard form kya hai?

Quadratic equation ka standard form ax² + bx + c = 0 hota hai, jahan a ≠ 0.

Q2. Discriminant kya batata hai?

Discriminant D = b² − 4ac roots ki nature batata hai. D positive hone par two distinct real roots, zero hone par equal real roots aur negative hone par non-real complex conjugate roots milte hain, jab coefficients real hon.

Q3. Quadratic equation ke roots ka sum kaise find karte hain?

Agar roots α aur β hain, toh α + β = −b/a.

Q4. Roots ka product kaise find karte hain?

Roots ka product αβ = c/a hota hai.

Q5. Equal roots ke liye kya condition hoti hai?

Equal roots ke liye discriminant zero hona chahiye: b² − 4ac = 0.

Q6. Agar roots diye hon toh quadratic equation kaise form karein?

Agar roots α aur β hain, toh monic quadratic equation x² − (α + β)x + αβ = 0 hogi.

Q7. Kya roots find kiye bina α² + β² calculate kar sakte hain?

Haan. α² + β² = (α + β)² − 2αβ use karo. Sum aur product coefficients se directly mil jaate hain.

Q8. JEE Main mein quadratic equations ke liye sabse useful shortcut kya hai?

Roots ko individually calculate karne se pehle sum, product aur discriminant ka use check karo. Bahut se questions inhi concepts se short ho jaate hain.

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Final Thoughts

Quadratic Equations mein strong command banane ke liye standard form, quadratic formula, discriminant aur roots-coefficients relations ko thoroughly revise karo. Uske baad sign of roots, equation formation aur parameter-based questions practice karo.

JEE Main ke questions solve karte waqt har baar roots directly calculate karne ki zaroorat nahi hoti. Agar question roots ke sum, product ya squares ke sum se related hai, toh Vieta's formulas aur algebraic identities ka use karo.

Remember: Discriminant roots ki nature batata hai, sum aur product roots ki properties batate hain, aur in concepts ka correct use aapko questions efficiently solve karne mein help karta hai.