Quadratic Equations
Nature of Roots
Grade 11
Question:
<p>If <strong>α</strong> and <strong>β</strong> (α ≠ β) are the roots of the equation <strong>x</strong><sup>2</sup> + <strong>bx</strong> + <strong>c</strong> = 0, where <strong>c</strong> ≠ 0 ≠ <strong>b</strong>, then</p>
<p>(a) 0 < α < β</p>
<p>(b) α < 0 < β < |α|</p>
<p>(c) α < β < 0</p>
<p>(d) α < 0 < |α| < β</p>
Step-by-Step Solution
Key Concept: Use the relationship between roots and coefficients along with the constraint that c ≠ 0 to determine the signs and ordering of the roots.
<p><strong>Step 1:</strong> We have α + β = −b and αβ = c</p><p><strong>Step 2:</strong> Since c ≠ 0, αβ ≠ 0, which means α ≠ 0 and β ≠ 0</p><p><strong>Step 3:</strong> Let α < 0 and β > 0</p><p><strong>Step 4:</strong> Since c ≠ 0 and c > 0 (implied), we have |α| = −α and α < 0 < β</p><p><strong>Step 5:</strong> From α + β = −b and β > 0, −|α| + β = −b, so β < |α|</p><p><strong>Step 6:</strong> Therefore, α < 0 < β < |α|</p><p>∴ Answer is (b)</p>
Correct Answer: b