<p>If the roots of the equation \(x^2 - 8x + a^2 - 6a = 0\) are real and distinct, then find all possible values of \(a\).</p>
Step-by-Step Solution
Key Concept: For real and distinct roots, the discriminant must be strictly positive (Δ > 0). Set up the inequality using Δ = b² - 4ac and solve the resulting quadratic inequality.
<p><strong>Step 1:</strong> For real and distinct roots, the discriminant must be strictly positive: Δ > 0</p><p><strong>Step 2:</strong> Calculate discriminant: Δ = b² - 4ac = (-8)² - 4(1)(a² - 6a) = 64 - 4a² + 24a</p><p><strong>Step 3:</strong> Set up inequality: 64 - 4a² + 24a > 0</p><p><strong>Step 4:</strong> Simplify by dividing by -4 (reverse inequality sign): a² - 6a - 16 < 0</p><p><strong>Step 5:</strong> Factor the quadratic: (a - 8)(a + 2) < 0</p><p><strong>Step 6:</strong> Using sign analysis or wavy curve method, the product is negative when one factor is positive and the other is negative, which occurs between the roots.</p><p><strong>Step 7:</strong> The roots are a = -2 and a = 8. Since the coefficient of a² is positive, the parabola opens upward, making the expression negative between the roots.</p><p>∴ Answer: <strong>-2 < a < 8</strong></p>
Correct Answer: -2 < a < 8