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Relationship Between Roots and Coefficients

Introduction

In quadratic equations, the coefficients and roots are connected by a simple relationship. This connection helps us calculate the sum and product of roots without solving the equation completely. It also allows us to form equations directly when roots are known.

Understanding this pattern saves time and builds intuition for higher-level algebra concepts.

Pattern: Relationship Between Roots and Coefficients

Pattern

For a quadratic equation ax² + bx + c = 0, if α and β are the roots:

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

These relations are derived from the factorized form of the quadratic equation:a(x - α)(x - β) = 0 → ax² - a(α + β)x + aαβ = 0.

Step-by-Step Example

Question

For the quadratic equation 3x² - 5x + 2 = 0, find the sum and product of its roots.

Solution

  1. Step 1: Identify coefficients

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

  2. Step 2: Apply formulas

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

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

  3. Step 3: Interpret

    The roots add up to 5/3 and multiply to 2/3.

  4. Final Answer:

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

  5. Quick Check:

    If roots are found by formula: x = (5 ± 1)/6 → 1 and 2/3 → Sum = 1.67, Product = 0.67 ✅

Quick Variations

1. Given sum and product, form a quadratic equation.

2. Find missing coefficient if one of the relationships is known.

3. Solve transformed equations when roots are increased, decreased, or doubled.

Trick to Always Use

  • Step 1: Identify a, b, c quickly from the equation.
  • Step 2: Use the direct formulas: α + β = -b/a, αβ = c/a.
  • Step 3: Keep track of negative signs carefully.

Summary

Summary

In the Relationship Between Roots and Coefficients pattern:

  • Sum of roots = -b/a, Product of roots = c/a.
  • Helps in finding relationships or forming equations without solving.
  • Always check signs to avoid common mistakes.

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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