Sorry for this question but I really have difficulties with model categories.

Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to H^*(A) $$ (Here $H^*(A)$ is the cohomology $dg$-algebra of $A$ with differential 0)

So my question is:

Is this equivalent to require, that there exists a $dg$-algebra $C$ and quasi-isomorphisms $$A\to C\leftarrow H^*(A) $$