If $X\subset \mathbb{P}^4$ is a 3-dimensional quadric hypersurface given by a rank 4 quadratic form (e.g. $X$ is a cone over a smooth quadric $\mathbb{P}^1\times\mathbb{P}^1\cong Q\subset\mathbb{P}^3$), then blowing up at the cone point $p\in X$ resolves the projection $\pi_p: X\dashrightarrow Q$ to express the blowup $Bl_pX\rightarrow Q$ as a $\mathbb{P}^1$ bundle over $Q$.
How do we see that $Bl_pX\cong \mathbb{P}(\mathscr{O}_{\mathbb{P}^1\times\mathbb{P}^1}\oplus \mathscr{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,1))$?
It is a special case of a more general result. If $Y \subset P^n$ is a projective variety and $X \subset P^{n+1}$ is the cone over $Y$ with vertex $p$, then $Bl_p(X) \cong P_Y(O_Y \oplus O_Y(-1))$.
To see this you can construct the maps between both sides by using the universal properties of the projectivization and of the blowup.