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 four distinct real roots is</p>
<p>(A) \((-\infty, 2)\)</p>
<p>(B) [incomplete in source]</p>
<p>(C) [incomplete in source]</p>
<p>(D) \((2, \infty)\)</p>
Step-by-Step Solution
Key Concept: Substitute \(y = x^2\) to reduce to quadratic. For four distinct real \(x\)-roots, need two distinct positive \(y\)-roots: discriminant > 0, sum > 0, product > 0.
<p>Let \(y = x^2\). Then the equation becomes \(y^2 - (k-1)y + (2-k) = 0\).</p><p>For four distinct real roots in \(x\), we need two distinct positive roots in \(y\).</p><p>Discriminant: \((k-1)^2 - 4(2-k) > 0\) ⟹ \(k^2 - 2k + 1 - 8 + 4k > 0\) ⟹ \(k^2 + 2k - 7 > 0\)</p><p>Sum of roots: \(k - 1 > 0\) ⟹ \(k > 1\)</p><p>Product of roots: \(2 - k > 0\) ⟹ \(k < 2\)</p><p>Combined: \(1 < k < 2\)</p>
Correct Answer: A