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Quadratic Equations (Standard Form)

Introduction

Quadratic equations algebra का एक fundamental हिस्सा हैं और इनका standard form होता है ax² + bx + c = 0 जहाँ a ≠ 0। ये aptitude और real-world problems-areas, projectile motion, optimization आदि में बहुत दिखते हैं।

इस pattern में mastery होने से आप equations को जल्दी factorization, completing the square, या quadratic formula से solve कर पाते हैं।

Pattern: Quadratic Equations (Standard Form)

Pattern

Key concept: Equation को ax² + bx + c = 0 के रूप में लिखकर, discriminant D = b² - 4ac निकालें और फिर factorization या quadratic formula का उपयोग करें:

x = [-b ± √(b² - 4ac)] / (2a)

Discriminant से roots का nature पता चलता है:

  • If D > 0 → दो अलग-अलग real roots मिलते हैं।
  • If D = 0 → एक repeated (equal) real root।
  • If D < 0 → दो complex conjugate roots।

Step-by-Step Example

Question

Quadratic equation solve करें: x² - 5x + 6 = 0.

Solution

  1. Step 1: Coefficients पहचानें

    a = 1, b = -5, c = 6.

  2. Step 2: Factorization try करें

    ऐसे दो numbers ढूँढें जिनका product = a·c = 6 और sum = b = -5 हो। ये हैं -2 और -3.

  3. Step 3: Middle term split करें

    x² - 5x + 6 = x² - 2x - 3x + 6.

  4. Step 4: Grouping और factoring

    (x² - 2x) - (3x - 6) = x(x - 2) - 3(x - 2) = (x - 2)(x - 3).

  5. Step 5: Zero-product property लागू करें

    (x - 2)(x - 3) = 0 ⇒ x - 2 = 0 या x - 3 = 0 ⇒ x = 2 या x = 3.

  6. Final Answer:

    x = 2, 3

  7. Quick Check:

    x = 2 रखने पर: 4 - 10 + 6 = 0 ✅. x = 3 रखने पर: 9 - 15 + 6 = 0 ✅.

Quick Variations

1. Coefficients fractions या negative हों तो LCM से multiply करके simplify करें।

2. Factorization मुश्किल हो तो quadratic formula use करें।

3. Completing the square vertex form निकालने या tough equations solve करने में मदद करता है।

4. Word problems (area, age, motion वगैरह) अक्सर quadratic में बदल जाते हैं।

Trick to Always Use

  • Step 1 → देखें कि quadratic आसानी से factor हो सकती है या नहीं (product = a·c, sum = b)।
  • Step 2 → Factor न हो तो discriminant D = b² - 4ac निकालकर आगे की method तय करें।
  • Step 3 → Quadratic formula का उपयोग करें और √D को ध्यान से simplify करें।
  • Step 4 → हर root को original equation में substitute करके quick-check करें।

Summary

Summary

Quadratic Equations (Standard Form) के key takeaways:

  • Solve करने से पहले equation को ax² + bx + c = 0 के form में लिखें।
  • Fast solving के लिए पहले factorization try करें; नहीं हो तो quadratic formula का उपयोग करें।
  • Roots के nature के लिए discriminant D = b² - 4ac चेक करें।
  • Sign mistakes से बचने के लिए हमेशा roots को substitute करके verify करें।

Practice

(1/5)
1. Solve: x² - 7x + 10 = 0
easy
A. 2 and 5
B. 3 and 4
C. 1 and 10
D. 5 and 10

Solution

  1. Step 1: Identify coefficients

    a = 1, b = -7, c = 10.

  2. Step 2: Factorise

    Find two numbers whose product = 10 and sum = -7 → -2 and -5.

  3. Step 3: Write factors

    (x - 2)(x - 5) = 0.

  4. Step 4: Solve for x

    x = 2 or x = 5.

  5. Final Answer:

    x = 2, 5 → Option A.

  6. Quick Check:

    Substitute x = 2 → 4 - 14 + 10 = 0 ✅

Hint: Find two numbers whose product is c and sum is b.
Common Mistakes: Choosing wrong factor pairs or sign errors.
2. Solve: x² - 9 = 0
easy
A. 9 and -9
B. 3 and -3
C. 1 and -1
D. 0 and 9

Solution

  1. Step 1: Recognise difference of squares

    x² - 9 = (x - 3)(x + 3).

  2. Step 2: Apply zero product rule

    (x - 3)(x + 3) = 0 ⇒ x = 3 or x = -3.

  3. Final Answer:

    x = 3, -3 → Option B.

  4. Quick Check:

    Substitute x = 3 → 9 - 9 = 0 ✅

Hint: Use a² - b² = (a - b)(a + b).
Common Mistakes: Forgetting the negative root or sign errors.
3. Solve: 2x² - 8x = 0
easy
A. 0 and 2
B. 2 and 4
C. 0 and 4
D. 4 and 8

Solution

  1. Step 1: Factor out common term

    2x² - 8x = 2x(x - 4).

  2. Step 2: Set each factor = 0

    2x = 0 ⇒ x = 0; x - 4 = 0 ⇒ x = 4.

  3. Final Answer:

    x = 0, 4 → Option C.

  4. Quick Check:

    Substitute x = 4 → 2(16) - 8(4) = 32 - 32 = 0 ✅

Hint: Take out the GCF (greatest common factor) before solving.
Common Mistakes: Missing the root x = 0 after factoring out x.
4. Solve: x² - 4x + 3 = 0
medium
A. 2 and 2
B. 4 and -1
C. 0 and 3
D. 1 and 3

Solution

  1. Step 1: Identify coefficients

    a = 1, b = -4, c = 3.

  2. Step 2: Find factor pair

    Product = 3, sum = -4 → -1 and -3.

  3. Step 3: Write factors

    (x - 1)(x - 3) = 0.

  4. Step 4: Solve for x

    x = 1 or x = 3.

  5. Final Answer:

    x = 1, 3 → Option D.

  6. Quick Check:

    Substitute x = 1 → 1 - 4 + 3 = 0 ✅

Hint: Look for two numbers whose product is c and sum is b.
Common Mistakes: Sign errors when choosing factor pairs.
5. Solve: x² + 2x - 8 = 0
medium
A. 2 and -4
B. 4 and -2
C. 3 and -2
D. 1 and -8

Solution

  1. Step 1: Identify coefficients

    a = 1, b = 2, c = -8.

  2. Step 2: Factorise

    Product = -8, sum = 2 → 4 and -2.

  3. Step 3: Write factors

    (x + 4)(x - 2) = 0.

  4. Step 4: Solve for x

    x = -4 or x = 2.

  5. Final Answer:

    x = 2, -4 → Option A.

  6. Quick Check:

    Substitute x = 2 → 4 + 4 - 8 = 0 ✅

Hint: When c is negative, look for factor pairs with opposite signs to get the required sum.
Common Mistakes: Incorrect sign assignment for factor pairs.

Mock Test

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