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

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$$
\left\{\begin{array}{l}
b \geqslant 0, \\
1+a+b \geqslant 0, \\
-\frac{a}{2} \in[0,1], \\
a^{2}-4 b \geqslant 0,
\end{array}\right. \text { draw the feasible region as shown in the figure. }
$$

When $z=a^{2}-2 b$ passes through the point $B(-2,1)$, $z_{\text {max }}=2$; when $z=a^{2}-2 b$ passes through the point $O(0,0)$, $z_{\min }=0$. Therefore, the range of $a^{2}-2 b$ is $[0,2]$.