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Forming Equation from Roots

Introduction

कई सवालों में आपको दिए गए roots (या roots के बीच संबंध) से एक quadratic equation बनानी होती है। यह pattern, standard relations का उपयोग करके root जानकारी को जल्दी से equation में बदलने में मदद करता है।

इस pattern में mastery होने से आप सीधे equations बना सकते हैं, बेवजह solving से बच सकते हैं, और transformed roots (जैसे α + k, 1/α) को भी आसानी से handle कर सकते हैं।

Pattern: Forming Equation from Roots

Pattern

मुख्य आइडिया: अगर α और β, ax² + bx + c = 0 (जहाँ a ≠ 0) के roots हैं, तो a(x - α)(x - β) expand होकर ax² - a(α + β)x + aαβ बनता है। यहाँ α + β = -b/a और αβ = c/a होते हैं।

Monic quadratic (a = 1) के लिए α और β वाले roots की equation होगी:
x² - (α + β)x + (αβ) = 0.

Step-by-Step Example

Question

वह monic quadratic equation बनाइए जिसके roots 2 और -3 हैं।

Solution

  1. Step 1: Roots का sum और product निकालें

    α + β = 2 + (-3) = -1. αβ = 2 × (-3) = -6.

  2. Step 2: Monic form में substitute करें

    Monic equation: x² - (α + β)x + (αβ) = 0 → x² - (-1)x + (-6) = 0.

  3. Step 3: Simplify करें

    x² + x - 6 = 0.

  4. Final Answer:

    x² + x - 6 = 0.

  5. Quick Check:

    (x + 3)(x - 2) ⇒ roots -3 और 2 आते हैं। Order matter नहीं करता ✅

Quick Variations

1. Non-monic equation: अगर general a चाहिए, तो monic form को a से multiply कर दें (fractions हटाने के लिए convenient integer चुनें)।

2. Transformed roots: अगर roots (α + k) और (β + k) हों, तो नया sum = (α + β) + 2k और product = αβ + k(α + β) + k² होगा।

3. Reciprocal roots: अगर roots 1/α और 1/β हों, तो sum = (α + β)/(αβ) और product = 1/(αβ) होगा। Equation उसी अनुसार बनाएं।

4. एक root direct हो और दूसरा relation में हो: अगर roots α और mα हों, तो sum = α(1 + m) और product = mα² होगा; दिए गए constraints से α को eliminate करें।

Trick to Always Use

  • Step 1 → हमेशा पहले roots का sum S और product P निकालें।
  • Step 2 → Monic equation के लिए x² - Sx + P = 0 इस्तेमाल करें। Fractions से बचने के लिए equation को LCM से multiply कर लें।
  • Step 3 → Transformed roots के लिए S और P को original α + β और αβ की मदद से algebraic identities लगाकर लिखें।

Summary

Summary

Forming Equation from Roots के लिए मुख्य बातें:

  • Required roots का sum S और product P निकालें।
  • Monic quadratic: x² - Sx + P = 0. Non-monic के लिए leading coefficient a से multiply करें।
  • Transformed roots (shift, reciprocal, scaling) के लिए नए S और P algebraic identities से निकालें।
  • Fractions हटाने के लिए equation को multiply करके साफ करें और हमेशा expand या factor करके quick-check करें।

Practice

(1/5)
1. Form the quadratic equation whose roots are 3 and 5.
easy
A. x² - 8x + 15 = 0
B. x² + 8x + 15 = 0
C. x² - 2x - 15 = 0
D. x² + 2x - 15 = 0

Solution

  1. Step 1: Find sum and product of roots

    Sum = 3 + 5 = 8. Product = 3 × 5 = 15.

  2. Step 2: Substitute in formula

    Monic equation: x² - (sum)x + (product) = 0 ⇒ x² - 8x + 15 = 0.

  3. Final Answer:

    x² - 8x + 15 = 0 → Option A.

  4. Quick Check:

    Expand (x - 3)(x - 5) = x² - 8x + 15 ✅

Hint: Equation = x² - (sum)x + (product).
Common Mistakes: Using wrong sign for the middle term (sign error on sum).
2. Form the quadratic equation whose roots are -2 and 4.
easy
A. x² + 2x - 8 = 0
B. x² - 2x - 8 = 0
C. x² - 2x + 8 = 0
D. x² + 2x + 8 = 0

Solution

  1. Step 1: Compute sum and product

    Sum = -2 + 4 = 2. Product = (-2) × 4 = -8.

  2. Step 2: Substitute

    x² - (sum)x + (product) = 0 ⇒ x² - 2x - 8 = 0.

  3. Final Answer:

    x² - 2x - 8 = 0 → Option B.

  4. Quick Check:

    (x + 2)(x - 4) = x² - 2x - 8 ✅

Hint: Sum = α + β, Product = αβ - plug into x² - Sx + P = 0.
Common Mistakes: Sign error for product when one root is negative.
3. Form the quadratic equation whose roots are 1/2 and 2/3.
easy
A. 6x² + 7x + 2 = 0
B. 6x² - 7x - 2 = 0
C. 6x² - 7x + 2 = 0
D. 3x² - 5x + 2 = 0

Solution

  1. Step 1: Sum and product

    Sum = 1/2 + 2/3 = (3 + 4)/6 = 7/6. Product = (1/2)(2/3) = 1/3.

  2. Step 2: Form fractional equation

    x² - (7/6)x + 1/3 = 0.

  3. Step 3: Clear denominators

    Multiply by 6 → 6x² - 7x + 2 = 0.

  4. Final Answer:

    6x² - 7x + 2 = 0 → Option C.

  5. Quick Check:

    Substitute x = 1/2 or 2/3 into 6x² - 7x + 2 to confirm ✅

Hint: Clear fractions by multiplying with LCM of denominators before finalizing.
Common Mistakes: Forgetting to multiply through to eliminate fractional coefficients.
4. If the roots are 2 and 1/3, form the quadratic equation.
medium
A. 3x² + 7x + 2 = 0
B. 3x² - 5x + 2 = 0
C. 3x² - 7x - 2 = 0
D. 3x² - 7x + 2 = 0

Solution

  1. Step 1: Compute sum and product

    Sum = 2 + 1/3 = 7/3. Product = 2 × 1/3 = 2/3.

  2. Step 2: Form equation

    x² - (7/3)x + (2/3) = 0.

  3. Step 3: Multiply through by 3

    3x² - 7x + 2 = 0.

  4. Final Answer:

    3x² - 7x + 2 = 0 → Option D.

  5. Quick Check:

    Plug x = 2 and x = 1/3 to confirm both satisfy the equation ✅

Hint: Use monic form then multiply to clear denominators for integer coefficients.
Common Mistakes: Arithmetic error when adding a whole number and a fraction.
5. Form the quadratic equation whose roots are -1/2 and -3/4.
medium
A. 8x² + 10x + 3 = 0
B. 8x² - 10x + 3 = 0
C. 4x² + 10x + 3 = 0
D. 4x² + 10x - 3 = 0

Solution

  1. Step 1: Find sum and product

    Sum = (-1/2) + (-3/4) = -5/4. Product = (-1/2)(-3/4) = 3/8.

  2. Step 2: Form monic equation

    x² - (sum)x + (product) = x² - (-5/4)x + 3/8 = x² + (5/4)x + 3/8.

  3. Step 3: Clear denominators

    Multiply by 8 → 8x² + 10x + 3 = 0.

  4. Final Answer:

    8x² + 10x + 3 = 0 → Option A.

  5. Quick Check:

    Substitute x = -1/2 and x = -3/4 to verify the equation holds ✅

Hint: If roots are fractions, clear denominators at the end by multiplying with LCM.
Common Mistakes: Dropping negative signs when summing negative fractions.

Mock Test

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