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 2 distinct real roots is</p>
<p>(A) \((0, 2)\)</p>
<p>(B) \((-\infty, -1)\)</p>
<p>(C) \((2, \infty)\)</p>
<p>(D) [incomplete in source]</p>
Step-by-Step Solution
Key Concept: Two distinct real \(x\)-roots occur when the quadratic in \(y\) has one positive and one negative root, requiring product < 0.
<p>For two distinct real roots in \(x\), either:</p><p>(i) One positive and one negative root of \(y\) (giving \(x = \pm\sqrt{y_1}\)), or</p><p>(ii) One root equals zero and one is negative (but negative \(y\) gives no real \(x\)).</p><p>Case (i): Product of roots < 0 ⟹ \(2 - k < 0\) ⟹ \(k > 2\)</p><p>Also need discriminant ≥ 0.</p><p>∴ Answer is \((2, \infty)\).</p>
Correct Answer: C