Step 1: Write the given relation.
We know $x+\dfrac{1}{x}=3$. We must find $x^2+\dfrac{1}{x^2}$.
Step 2: Pick the right identity.
Squaring a sum gives $\left(x+\dfrac{1}{x}\right)^2=x^2+\dfrac{1}{x^2}+2$, because the middle term $2\cdot x\cdot\dfrac{1}{x}=2$.
Step 3: Square the given value.
Since $x+\dfrac{1}{x}=3$, squaring both sides gives \[ \left(x+\frac{1}{x}\right)^2=3^2=9. \]
Step 4: Expand the left side.
Using the identity, \[ x^2+\frac{1}{x^2}+2=9. \]
Step 5: Move the $2$ across.
Subtract $2$ from both sides. \[ x^2+\frac{1}{x^2}=9-2=7. \]
Step 6: State the answer.
So the required value is $7$. We never needed to actually solve for $x$. \[ \boxed{7} \]