Question:

<p><strong>Step 1:</strong> For <span class="math-tex">\(x^2 + ax + b = 0\)</span>, let roots be <span class="math-tex">\(r_1, r_2\)</span>. Difference = <span class="math-tex">\(|r_1 - r_2| = \sqrt{a^2 - 4b}\)</span></p><p><strong>Step 2:</strong> For <span class="math-tex">\(x^2 + bx + a = 0\)</span>, let roots be <span class="math-tex">\(s_1, s_2\)</span>. Difference = <span class="math-tex">\(|s_1 - s_2| = \sqrt{b^2 - 4a}\)</span></p><p><strong>Step 3:</strong> Given that the differences are equal: <span class="math-tex">\(\sqrt{a^2 - 4b} = \sqrt{b^2 - 4a}\)</span></p><p><strong>Step 4:</strong> Squaring both sides: <span class="math-tex">\(a^2 - 4b = b^2 - 4a\)</span></p><p><strong>Step 5:</strong> <span class="math-tex">\(a^2 - b^2 = 4b - 4a\)</span></p><p><strong>Step 6:</strong> <span class="math-tex">\((a-b)(a+b) = -4(a-b)\)</span></p><p><strong>Step 7:</strong> Since <span class="math-tex">\(a \neq b\)</span>, we can divide by <span class="math-tex">\((a-b)\)</span>: <span class="math-tex">\(a + b = -4\)</span></p><p>∴ Answer is (A).</p>