<p>If one root of the equation \(z^2 - az + a - 1 = 0\) is \((1+i)\), where \(a\) is a complex number, find the other root.</p>
Step-by-Step Solution
Key Concept: For a quadratic equation with complex coefficients, use Vieta's formulas (sum and product of roots) directly—the conjugate root theorem does NOT apply unless coefficients are real. Here, substitute the given root to find 'a', then use Vieta's formulas to find the other root.
<p><strong>Step 1:</strong> Substitute z = (1+i) into the equation to find 'a'.</p><p>(1+i)² − a(1+i) + a − 1 = 0</p><p>1 + 2i − 1 − a(1+i) + a − 1 = 0</p><p>2i − a − ai + a − 1 = 0</p><p>2i − ai − 1 = 0</p><p>a(1+i) = 2i − 1</p><p>a = (−1+2i)/(1+i)</p><p><strong>Step 2:</strong> Rationalize by multiplying by (1−i)/(1−i).</p><p>a = (−1+2i)(1−i)/[(1+i)(1−i)] = (−1+i+2i−2i²)/(1+1)</p><p>a = (−1+3i+2)/2 = (1+3i)/2</p><p><strong>Step 3:</strong> Use Vieta's formula: sum of roots = a.</p><p>(1+i) + z₂ = (1+3i)/2</p><p>z₂ = (1+3i)/2 − (1+i) = (1+3i−2−2i)/2</p><p>z₂ = (−1+i)/2</p><p>∴ <strong>The other root is z = (−1+i)/2</strong></p>
Correct Answer: z = 1