Quadratic Equations
Nature of Roots
Grade 11
Question:
<p>If a, b, c are real and <span class="math">a \neq b</span>, the roots of the equation <span class="math">2(a - b)x^2 - 11(a + b + c)x - 3(a - b) = 0</span> are</p>
<p>(a) real and equal</p>
<p>(b) real and unequal</p>
<p>(c) purely imaginary</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: To determine the nature of roots of a quadratic equation, we compute the discriminant (Δ = b² - 4ac). If Δ > 0, roots are real and unequal; if Δ = 0, roots are real and equal; if Δ < 0, roots are complex. We must verify that Δ is strictly positive given the constraints a ≠ b.
<p><strong>Step 1:</strong> Identify the quadratic equation coefficients.</p><p>The equation is: 2(a - b)x² - 11(a + b + c)x - 3(a - b) = 0</p><p>Here: A = 2(a - b), B = -11(a + b + c), C = -3(a - b)</p><p><strong>Step 2:</strong> Calculate the discriminant Δ = B² - 4AC.</p><p>Δ = [-11(a + b + c)]² - 4[2(a - b)][-3(a - b)]</p><p>Δ = 121(a + b + c)² + 24(a - b)²</p><p><strong>Step 3:</strong> Analyze the sign of the discriminant.</p><p>Since a, b, c are real numbers:</p><p>• 121(a + b + c)² ≥ 0 (perfect square term)</p><p>• 24(a - b)² > 0 (since a ≠ b, this term is strictly positive)</p><p>Therefore: Δ = 121(a + b + c)² + 24(a - b)² > 0</p><p><strong>Step 4:</strong> Determine the nature of roots.</p><p>Since Δ > 0, the quadratic equation has two distinct real roots that are unequal.</p><p><strong>∴ Answer:</strong> B (real and unequal)</p>
Correct Answer: B