Quadratic Equations
Roots and Their Properties
Grade 11
Question:
<p>We have, <span class="math">\((5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0\)</span></p><p>Find the harmonic mean of the roots.</p>
Step-by-Step Solution
Key Concept: The harmonic mean of roots is calculated as 2×(product of roots)/(sum of roots). Simplify by rationalizing if needed.
<p><strong>Step 1:</strong> Sum of the roots $= \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$</p><p><strong>Step 2:</strong> Product of the roots $= \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}}$</p><p><strong>Step 3:</strong> Harmonic mean of the roots $= \frac{2 \times \text{Product of roots}}{\text{Sum of roots}} = \frac{2 \times (8 + 2\sqrt{5})}{4 + \sqrt{5}} = 4$</p>
Correct Answer: 4