Quadratic Equations
Properties of Roots
Grade 11
Question:
<p>If the difference between the roots of <span class="inline-math">x^2 + ax + b = 0\</span> is same as that of <span class="inline-math">x^2 + bx + a = 0\</span>, <span class="inline-math">a \neq b\</span>, then:</p>
<p>(a) <span class="inline-math">a + b + 4 = 0\</span></p>
<p>(b) <span class="inline-math">a + b - 4 = 0\</span></p>
<p>(c) <span class="inline-math">a - b - 4 = 0\</span></p>
<p>(d) <span class="inline-math">a - b + 4 = 0\</span></p>
Step-by-Step Solution
Key Concept: The difference between roots of a quadratic x² + px + q = 0 is √(p² - 4q). Since this difference is the same for both equations, we equate the expressions and simplify using the constraint a ≠ b.
<p><strong>Step 1:</strong> For equation x² + ax + b = 0, let roots be α and β. By Vieta's formulas: α + β = -a and αβ = b. The difference between roots is |α - β| = √((α + β)² - 4αβ) = √(a² - 4b)</p><p><strong>Step 2:</strong> For equation x² + bx + a = 0, let roots be γ and δ. Similarly: γ + δ = -b and γδ = a. The difference between roots is |γ - δ| = √(b² - 4a)</p><p><strong>Step 3:</strong> Given that the differences are equal: √(a² - 4b) = √(b² - 4a)</p><p><strong>Step 4:</strong> Squaring both sides: a² - 4b = b² - 4a</p><p><strong>Step 5:</strong> Rearranging: a² - b² - 4b + 4a = 0</p><p><strong>Step 6:</strong> Factoring: (a² - b²) + 4(a - b) = 0 → (a - b)(a + b) + 4(a - b) = 0</p><p><strong>Step 7:</strong> Factoring out (a - b): (a - b)(a + b + 4) = 0</p><p><strong>Step 8:</strong> This gives either a - b = 0 or a + b + 4 = 0. Since we are given a ≠ b, we must have a + b + 4 = 0</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A