<p>Let \(x^2 - ax + 30 = y\) and \(y = 2\sqrt{y} + 15\). Given that \(\lambda = \frac{\alpha + \beta}{2}\) where \(\alpha, \beta\) are roots of \(x^2 - ax + 20 = 0\). Find the minimum value of \(\lambda\).</p>
Step-by-Step Solution
Key Concept: Reduce the equation system to a quadratic, then apply AM-GM inequality to find the minimum value of the arithmetic mean.
<p><strong>Step 1:</strong> From $y = 2\sqrt{y} + 15$, we get $y^2 - 4y - 60 = 0$</p><p><strong>Step 2:</strong> Solving: $(y - 10)(y + 6) = 0$, so $y = 10$ (since $y > 0$)</p><p><strong>Step 3:</strong> From $x^2 - ax + 30 = 10$, we get $x^2 - ax + 20 = 0$</p><p><strong>Step 4:</strong> For roots $\alpha, \beta$: $\alpha\beta = 20$</p><p><strong>Step 5:</strong> By AM-GM inequality: $\frac{\alpha + \beta}{2} \geq \sqrt{\alpha\beta} = \sqrt{20} = 2\sqrt{5}$</p><p><strong>Step 6:</strong> But $\lambda = \frac{\alpha + \beta}{2}$, so minimum $\lambda = 4\sqrt{5}$ (when $\mu = 4\sqrt{5} \approx 8.9$)</p>
Correct Answer: \(4\sqrt{5}\)