Question

The roots of the equation $2 (a ^ 2 + b ^ 2) \times x ^ 2 + 2(a + b) \times x + 1 = 0$ are

• A. Real and equal
• B. Real and irrational
• C. Imaginary
• D. None of the above

Answer 1

Answer:

C. Imaginary

Understanding the Nature of Roots of a Quadratic Equation

• The given equation is in the standard quadratic form: Ax² + Bx + C = 0.
• To determine the nature of the roots of a quadratic equation, we calculate its discriminant, denoted by Δ (delta) or D.
• The formula for the discriminant is Δ = B² - 4AC.

Interpreting the Discriminant

• If Δ > 0: The roots are real and distinct (unequal).
• If Δ = 0: The roots are real and equal.
• If Δ < 0: The roots are imaginary (complex conjugates).

Applying to the Given Equation

1. First, identify the coefficients A, B, and C from the given equation: 2 (a² + b²) x² + 2(a + b) x + 1 = 0.
   • A = 2(a² + b²)
   • B = 2(a + b)
   • C = 1
2. Next, substitute these values into the discriminant formula:Δ = B² - 4ACΔ = (2(a + b))² - 4 × [2(a² + b²)] × 1
3. Simplify the expression:Δ = 4(a + b)² - 8(a² + b²)Δ = 4(a² + 2ab + b²) - 8a² - 8b²Δ = 4a² + 8ab + 4b² - 8a² - 8b²Δ = -4a² + 8ab - 4b²
4. Factor out -4:Δ = -4(a² - 2ab + b²)
5. Recognize the perfect square trinomial:Δ = -4(a - b)²

Conclusion on the Nature of Roots

• For any real numbers 'a' and 'b', the term (a - b)² is always greater than or equal to zero ((a - b)² ≥ 0).
• Therefore, -4(a - b)² will always be less than or equal to zero (Δ ≤ 0).
• This implies two possibilities for the roots:
  • If a = b, then (a - b)² = 0, which makes Δ = 0. In this specific case, the roots would be real and equal.
  • If a ≠ b, then (a - b)² > 0, which makes Δ < 0 (a negative value). In this general case, the roots are imaginary.
• Since 'Imaginary' is provided as the correct answer, it implies that the question considers the general scenario where a ≠ b, leading to a strictly negative discriminant.

Key Points for Competitive Exams

• Always be prepared to calculate the discriminant for various types of quadratic equations.
• Remember the three conditions (Δ > 0, Δ = 0, Δ < 0) and their corresponding root natures.
• Pay close attention to algebraic manipulations, especially factoring perfect squares, as they simplify complex discriminant expressions.
• In multiple-choice questions, if a general answer like 'Imaginary' is given, it usually refers to the most common or general case where parameters are not equal, leading to that specific nature of roots.