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Relationship Between Roots and Coefficients

Introduction

In quadratic equations-ல், coefficients மற்றும் roots இடையே ஒரு எளிய தொடர்பு உள்ளது. இந்த தொடர்பு, சமன்பாட்டை முழுமையாக தீர்க்காமல் roots-ன் sum மற்றும் product-ஐ கணக்கிட உதவுகிறது. மேலும், roots கொடுக்கப்பட்டால் நேரடியாக புதிய equations உருவாக்கவும் இது பயன்படும்.

இந்த pattern-ஐ புரிந்துகொண்டால் நேரம் சேமிக்க முடியும், மேலும் higher-level algebra concepts-க்கு தேவையான intuition உருவாகும்.

Pattern: Relationship Between Roots and Coefficients

Pattern

ax² + bx + c = 0 என்ற quadratic equation-க்கு, α மற்றும் β roots ஆக இருந்தால்:

  • roots-ன் கூட்டுத்தொகை (α + β) = -b/a
  • roots-ன் பெருக்கல் (αβ) = c/a

இந்த தொடர்புகள் quadratic equation-ன் factorized form-இலிருந்து பெறப்பட்டவை: a(x - α)(x - β) = 0 → ax² - a(α + β)x + aαβ = 0.

Step-by-Step Example

Question

3x² - 5x + 2 = 0 என்ற quadratic equation-க்கு, roots-ன் sum மற்றும் product-ஐ கண்டறியவும்.

Solution

  1. Step 1: Identify coefficients

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

  2. Step 2: Apply formulas

    roots-ன் sum (α + β) = -b/a = -(-5)/3 = 5/3.

    roots-ன் product (αβ) = c/a = 2/3.

  3. Step 3: Interpret

    roots-ன் கூட்டுத்தொகை 5/3 ஆகவும், பெருக்கல் 2/3 ஆகவும் உள்ளது.

  4. Final Answer:

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

  5. Quick Check:

    formula மூலம் roots-ஐ கண்டால்: x = (5 ± 1)/6 → 1 மற்றும் 2/3 → Sum = 1.67, Product = 0.67 ✅

Quick Variations

1. sum மற்றும் product கொடுக்கப்பட்டால், quadratic equation-ஐ உருவாக்குதல்.

2. ஒரு relationship தெரிந்தால் missing coefficient-ஐ கண்டறிதல்.

3. roots அதிகரிக்கப்பட்டால், குறைக்கப்பட்டால், அல்லது double செய்யப்பட்டால் வரும் transformed equations-ஐ தீர்க்குதல்.

Trick to Always Use

  • Step 1: equation-இலிருந்து a, b, c-ஐ விரைவாக கண்டறியுங்கள்.
  • Step 2: நேரடி formulas பயன்படுத்துங்கள்: α + β = -b/a, αβ = c/a.
  • Step 3: negative signs-ஐ கவனமாக கையாளுங்கள்.

Summary

Summary

Relationship Between Roots and Coefficients pattern-ல்:

  • roots-ன் sum = -b/a, roots-ன் product = c/a.
  • equation-ஐ தீர்க்காமல் relationships கண்டறியவும், equations உருவாக்கவும் உதவும்.
  • பொதுவான தவறுகளை தவிர்க்க signs-ஐ எப்போதும் சரிபார்க்கவும்.

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.

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