1. **Stating the problem:** We have two curves representing pressure $P$ as a function of flow rate $Q$: - System curve: $P = a Q^2 - b Q + c$ - Ventilator curve: $P = -7 \times 10^{-8} Q^2 + 3 \times 10^{-4} Q + 19.6$ We want to find the intersection point $w$ where these two curves meet, i.e., where their $P$ values are equal for the same $Q$. 2. **Set the two equations equal to find $Q$ at the intersection:** $$a Q^2 - b Q + c = -7 \times 10^{-8} Q^2 + 3 \times 10^{-4} Q + 19.6$$ 3. **Rearrange to form a quadratic equation:** $$a Q^2 - b Q + c + 7 \times 10^{-8} Q^2 - 3 \times 10^{-4} Q - 19.6 = 0$$ Combine like terms: $$\left(a + 7 \times 10^{-8}\right) Q^2 - \left(b + 3 \times 10^{-4}\right) Q + (c - 19.6) = 0$$ 4. **Use the quadratic formula to solve for $Q$:** $$Q = \frac{\left(b + 3 \times 10^{-4}\right) \pm \sqrt{\left(b + 3 \times 10^{-4}\right)^2 - 4 \left(a + 7 \times 10^{-8}\right) (c - 19.6)}}{2 \left(a + 7 \times 10^{-8}\right)}$$ 5. **Interpretation:** The values of $a$, $b$, and $c$ must be known to compute the exact intersection points. Once $Q$ is found, substitute back into either equation to find $P$. 6. **Summary:** The intersection point $w$ is the solution $(Q, P)$ satisfying both curves simultaneously, found by solving the quadratic equation above. If you provide values for $a$, $b$, and $c$, I can compute the exact intersection point.