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Relationship Between Roots and Coefficients

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Introduction

Quadratic equations में coefficients और roots एक simple relationship से जुड़े होते हैं। यह connection हमें बिना equation को पूरी तरह solve किए roots का sum और product निकालने में मदद करता है। यह हमें तब भी equation बनाने देता है जब roots पहले से दिए हों।

इस pattern को समझना समय बचाता है और higher-level algebra concepts के लिए intuition बढ़ाता है।

Pattern: Relationship Between Roots and Coefficients

Pattern: Relationship Between Roots and Coefficients

एक quadratic equation ax² + bx + c = 0 में, अगर α और β roots हों:

  • Sum of roots (α + β) = -b/a
  • Product of roots (αβ) = c/a

ये relations quadratic equation के factorized form से मिलती हैं: a(x - α)(x - β) = 0 → ax² - a(α + β)x + aαβ = 0.

Step-by-Step Example

Question

Quadratic equation 3x² - 5x + 2 = 0 के लिए, इसके roots का sum और product निकालें।

Solution

  1. Step 1: Coefficients पहचानें

    a = 3, b = -5, c = 2.

  2. Step 2: Formulas apply करें

    Sum of roots (α + β) = -b/a = -(-5)/3 = 5/3.

    Product of roots (αβ) = c/a = 2/3.

  3. Step 3: Interpretation

    Roots का sum 5/3 और product 2/3 है।

  4. Final Answer:

    Sum = 5/3, Product = 2/3.

  5. Quick Check:

    अगर roots को formula से निकालें: x = (5 ± 1)/6 → 1 और 2/3 → Sum = 1.67, Product = 0.67 ✅

Quick Variations

1. Sum और product दिए हों तो quadratic equation बनाएं।

2. अगर relation दिया हो तो missing coefficient निकालें।

3. जब roots बढ़ाए, घटाए या double किए जाएँ, तो नई transformed equation solve करें।

Trick to Always Use

  • Step 1: Equation से a, b, c तुरंत पहचानें।
  • Step 2: Direct formulas यूज़ करें: α + β = -b/a, αβ = c/a.
  • Step 3: Negative signs को ध्यान से देखें।

Summary

Relationship Between Roots and Coefficients pattern में:

  • Sum of roots = -b/a, Product of roots = c/a.
  • यह equations को बिना solve किए relationships बनाने में मदद करता है।
  • गलतियों से बचने के लिए signs हमेशा चेक करें।

Practice

(1/5)
1. Find the sum and product of the roots of the equation x² + 7x + 10 = 0.
easy
A. Sum = -7, Product = 10
B. Sum = 7, Product = 10
C. Sum = -7, Product = -10
D. Sum = 7, Product = -10

Solution

  1. Step 1: Identify coefficients

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

  2. Step 2: Apply formulas

    Sum = -b/a = -7/1 = -7.

    Product = c/a = 10/1 = 10.

  3. Final Answer:

    Sum = -7, Product = 10 → Option A.

  4. Quick Check:

    Factors: (x + 5)(x + 2) ⇒ roots -5 and -2 → Sum = -7, Product = 10 ✅

Hint: Use Sum = -b/a and Product = c/a directly.
Common Mistakes: Forgetting the negative sign in -b/a or misreading coefficients.
2. For the equation 2x² - 5x + 2 = 0, find the sum and product of roots.
easy
A. Sum = -5/2, Product = 2
B. Sum = -5/2, Product = 1
C. Sum = 5/2, Product = 1
D. Sum = 5/2, Product = 2

Solution

  1. Step 1: Identify coefficients

    a = 2, b = -5, c = 2.

  2. Step 2: Apply formulas

    Sum = -b/a = -(-5)/2 = 5/2.

    Product = c/a = 2/2 = 1.

  3. Final Answer:

    Sum = 5/2, Product = 1 → Option C.

  4. Quick Check:

    Factorisation: (2x - 1)(x - 2) ⇒ roots 1/2 and 2 → Sum = 1/2 + 2 = 5/2, Product = 1/2 × 2 = 1 ✅

Hint: Carefully note the sign of b; apply -b/a and c/a after identifying a, b, c.
Common Mistakes: Confusing the sign of b or dividing c by a incorrectly.
3. If the roots of 3x² + 2x - 1 = 0 are α and β, find α + β and αβ.
easy
A. α + β = -2/3, αβ = -1/3
B. α + β = 2/3, αβ = -1/3
C. α + β = -2/3, αβ = 1/3
D. α + β = 2/3, αβ = 1/3

Solution

  1. Step 1: Identify coefficients

    a = 3, b = 2, c = -1.

  2. Step 2: Apply formulas

    Sum = -b/a = -2/3.

    Product = c/a = -1/3.

  3. Final Answer:

    α + β = -2/3, αβ = -1/3 → Option A.

  4. Quick Check:

    Factorisation: (3x - 1)(x + 1) ⇒ roots 1/3 and -1 → Sum = 1/3 - 1 = -2/3, Product = -1/3 ✅

Hint: Divide b and c by a first if that helps to see fractions clearly.
Common Mistakes: Sign errors for c/a when c is negative.
4. For the equation 4x² + 3x + 2 = 0, find α + β and αβ.
medium
A. α + β = -3/4, αβ = 1/2
B. α + β = 3/4, αβ = 1/2
C. α + β = -3/4, αβ = 3/4
D. α + β = 3/4, αβ = -1/2

Solution

  1. Step 1: Coefficients

    a = 4, b = 3, c = 2.

  2. Step 2: Use formulas

    Sum = -b/a = -3/4.

    Product = c/a = 2/4 = 1/2.

  3. Final Answer:

    α + β = -3/4, αβ = 1/2 → Option A.

  4. Quick Check:

    Sum and product computed directly from coefficients: -3/4 and 1/2 - consistent with formulas ✅

Hint: If a ≠ 1, remember to divide b and c by a when thinking in terms of monic polynomial.
Common Mistakes: Forgetting to divide by a when computing product c/a.
5. If the roots of 5x² - 6x + 1 = 0 are α and β, find α + β and αβ.
medium
A. α + β = -6/5, αβ = 1/5
B. α + β = -6/5, αβ = -1/5
C. α + β = 6/5, αβ = -1/5
D. α + β = 6/5, αβ = 1/5

Solution

  1. Step 1: Identify coefficients

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

  2. Step 2: Compute

    Sum = -b/a = -(-6)/5 = 6/5.

    Product = c/a = 1/5.

  3. Final Answer:

    α + β = 6/5, αβ = 1/5 → Option D.

  4. Quick Check:

    Factorisation: (5x - 1)(x - 1) ⇒ roots 1/5 and 1 → Sum = 1/5 + 1 = 6/5, Product = 1/5 ✅

Hint: When b is negative, -b/a becomes positive - watch the signs.
Common Mistakes: Confusing sign of -b when b is already negative.