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  4. If the equation x2-4 x+ log (1/2)(a)=0 does not have distinct real roots, then find the minimum value of (1/a).

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Q. If the equation $x^2-4 x+\log _{\frac{1}{2}}(a)=0$ does not have distinct real roots, then find the minimum value of $\frac{1}{a}$.

Complex Numbers and Quadratic Equations

Solution:

$x^2-4 x-\log _2 a=0$
$D \leq 0 $
$16+4 \log _2 a \leq 0 \Rightarrow 4+\log _2 a \leq 0 \Rightarrow \log _2 a \leq-4 \Rightarrow a \leq \frac{1}{16} $
$\left. a _{\text {max. }}=\frac{1}{16} \Rightarrow \frac{1}{ a }\right]_{\min .}=16$