Quadratic Equations
Biquadratic Equations
Grade 11
Question:
<p>Consider the equation \(x^4 - (k-1)x^2 + (2-k) = 0\). The complete set of possible values of real \(k\) for which the equation has 3 distinct real roots is</p>
<p>(A) \(\{2\}\)</p>
<p>(B) \(\{2-\sqrt{1}, 2\}\)</p>
<p>(C) [incomplete in source]</p>
<p>(D) [incomplete in source]</p>
Step-by-Step Solution
Key Concept: Three distinct real roots occur when the biquadratic has one zero root (with multiplicity 2) and one positive root.
<p>For three distinct real roots, one root of \(x^2 = 0\) (giving \(x = 0\) with multiplicity 2) and one positive root.</p><p>This occurs when one root of \(y^2 - (k-1)y + (2-k) = 0\) is \(y = 0\).</p><p>Substituting \(y = 0\): \(2 - k = 0\) ⟹ \(k = 2\)</p><p>When \(k = 2\): equation becomes \(y^2 - y = 0\) ⟹ \(y = 0\) or \(y = 1\)</p><p>This gives \(x = 0\) (double root) and \(x = \pm 1\) (distinct roots).</p><p>∴ Answer is \(k = 2\).</p>
Correct Answer: A