Example 2

The demand function of a good is given by $q_d(p)=\frac{1}{p}$, $(p>0)$ and the supply function by $q_s(p)=3p+2$, $(p\geq 0)$. We determine the market equilibrium.

This implies the following:
$$\begin{align} q_d(p)=q_s(p)& \Leftrightarrow \frac{1}{p}=3p+2\\ & \Leftrightarrow 1=3p^2+2p\\ & \Leftrightarrow 3p^2+2p-1=0\\ & \Leftrightarrow p=-1 \mbox { or } p=\frac{1}{3}, \end{align}$$

where we used the quadratic formula in the final step.

Due to non-negativity $p=-1$ is not allowed. Hence, the market equilibrium is $(q,p)=(3,\frac{1}{3})$.