<p><strong>Ex. 25 (A):</strong> If <p>a + b + 2c = 0</p> but <p>c ≠ 0</p>, then the equation <p>ax² + bx + c = 0</p> has at least one root in which interval?</p>
Step-by-Step Solution
Key Concept: Use the Intermediate Value Theorem: if f(0)·f(1) < 0, then there exists a root between 0 and 1.
<p><strong>Step 1:</strong> Let <p>f(x) = ax² + bx + c</p></p><p><strong>Step 2:</strong> Then <p>f(1) = a + b + c = -c</p> [since <p>a + b + 2c = 0</p>, so <p>a + b = -2c</p>]</p><p><strong>Step 3:</strong> And <p>f(0) = c</p></p><p><strong>Step 4:</strong> Therefore <p>f(0)·f(1) = -c² < 0</p> [since <p>c ≠ 0</p>]</p><p><strong>Step 5:</strong> By Intermediate Value Theorem, <p>f(x) = 0</p> has a root in <p>(0, 1)</p></p><p><strong>Step 6:</strong> The equation <p>f(x) = 0</p> has roots in <p>(-2, 0)</p> (p) and <p>(-1, 1)</p> (r)</p>
Correct Answer: p, r