Quadratic Equations
Nature of Roots
Grade 11
Question:
<p>If <span>\(a \neq b\)</span>, then the roots of the equation <span>\(2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0\)</span> are</p>
<p>(a) real and distinct</p>
<p>(b) real and equal</p>
<p>(c) imaginary</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: When the discriminant of a quadratic equation is negative, the roots are complex conjugates (imaginary).
<p><strong>Solution:</strong></p><p>The given equation is <span>$2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0$</span></p><p>The discriminant is:</p><p><span>$D = 4(a + b)^2 - 8(a^2 + b^2)$</span></p><p><span>$= 4(a^2 + 2ab + b^2) - 8(a^2 + b^2)$</span></p><p><span>$= 4a^2 + 8ab + 4b^2 - 8a^2 - 8b^2$</span></p><p><span>$= -4a^2 + 8ab - 4b^2$</span></p><p><span>$= -4(a^2 - 2ab + b^2)$</span></p><p><span>$= -4(a - b)^2 < 0$</span> (since <span>$a \neq b$</span>)</p><p>Since <span>$D < 0$</span>, the roots are imaginary.</p><p>∴ Answer is (c).</p>
Correct Answer: C