No fake roots allowed! :P

As the roots of the given equation are real,the discriminant of the equation $\geq 0$

$16+4log_{2}{a} \geq 0$

$=> log_{2}{a} \geq - 4$

$=> log_{2}{a} \geq log_{2}{2}^{-4}$

$=>a \geq 2^{-4}$

So, $\boxed{a \geq \frac{1}{16}}$

Let us call this polynomial P(x) for convenience,

P(x) has no real roots <=> for all real x, P(x) > 0

the negative of this statement is

P(x) has at least one real root <=> there exist x real such that P(x) $\le$ 0

If x has no real roots then ${ x }^{ 2 }-4x-\log _{ 2 }{ a } > 0$ This implies that for all real x,

${ 2 }^{ x(x-4) } > a$

Since the left handside cannot be smaller than ${ 2 }^{ -4 }$ , a must be strictly smaller than ${ 2 }^{ -4 }$ for P(x) to have no real roots.

We have shown that: for all real x, P(x) > 0 <=> a < ${ 2 }^{ -4 }$

The negative of this is: there exist x real such that P(x) $\le$ 0 <=> a $\ge$ ${2 }^{ -4 }$

If that statement is true then P(x) has at least one real root.

So, for P(x) to have real roots, a must be bigger or equal to $\boxed { { 2 }^{ -4 }\quad or\quad \frac { 1 }{ 16 }}$