Quadratic Equations
Nature of Roots
Grade None
Question:
<p>For the equation \(2x^2 + 6\sqrt{2}x + 1 = 0\)</p>
<p>(a) roots are rational</p>
<p>(b) roots are irrational</p>
<p>(c) if one root is \(p + \sqrt{q}\), the other is \(-p + \sqrt{q}\)</p>
<p>(d) if one root is \(p + \sqrt{q}\), the other is \(p - \sqrt{q}\)</p>
Step-by-Step Solution
Key Concept: When coefficients are irrational, conjugate pairs don't follow the rational coefficient rule. Roots can be irrational without being conjugate pairs.
<p><strong>Solution:</strong> As the coefficients are not rational, irrational roots need not appear in conjugate pair.</p><p>Let $\alpha = p + \sqrt{q}$, then prove that other root $\beta = -p + \sqrt{q}$.</p><p>∴ The correct options are (b) and (c).</p>
Correct Answer: b,c