3. Let $f(x)=x^{2}+a x+b$ have two real roots in the interval $[0,1]$. Then the range of $a^{2}-2 b$ is
3. $[0,2]$.

Notice that, $f(x)=\left(x+\frac{a}{2}\right)^{2}+b-\frac{a^{2}}{4}$ has two real roots in the interval $[0,1]$.
According to the problem, $a$ and $b$ satisfy
$$
\begin{array}{l}
f(0)=b \geqslant 0, f(1)=a+b+1 \geqslant 0, \\
a^{2}-4 b \geqslant 0,0 \leqslant-\frac{a}{2} \leqslant 1 .
\end{array}
$$

In summary, the range of $a^{2}-2 b$ is $[0,2]$.