Question

If the roots of the quadratic equation $x^{2}-6x+log_{2}\lambda =0$ are real, then find the maximum value of $\lambda $

Answer 1

For real roots,
discriminant $\geq 0$
$\Leftrightarrow 36-4log_{2}\lambda \geq 0$
$\Leftrightarrow 4log_{2}\lambda -36\leq 0$
$\Leftrightarrow log_{2}\lambda \leq 9$
$\Leftrightarrow \lambda \leq 2^{9}$
$\Leftrightarrow \lambda \leq 512$
$\Rightarrow $ Maximum value of $\lambda =512$