1. **State the problem:** We are given demand and supply functions for a commodity: Demand: $P_d = 18 - 2x - x^2$ Supply: $P_s = 2x - 3$ We need to find the consumer's surplus and producer's surplus at the equilibrium price. 2. **Find the equilibrium quantity and price:** At equilibrium, demand equals supply: $$18 - 2x - x^2 = 2x - 3$$ Rearranging: $$18 + 3 = 2x + 2x + x^2$$ $$21 = 4x + x^2$$ Rewrite as: $$x^2 + 4x - 21 = 0$$ 3. **Solve the quadratic equation:** Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=4$, $c=-21$: $$x = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2}$$ Possible solutions: $$x = \frac{-4 + 10}{2} = 3$$ $$x = \frac{-4 - 10}{2} = -7$$ Since quantity cannot be negative, equilibrium quantity is $x=3$. 4. **Find equilibrium price:** Substitute $x=3$ into either function, e.g., supply: $$P = 2(3) - 3 = 6 - 3 = 3$$ 5. **Calculate consumer's surplus:** Consumer surplus is the area between the demand curve and the equilibrium price from 0 to equilibrium quantity: $$CS = \int_0^3 (P_d - P_e) \, dx = \int_0^3 \left(18 - 2x - x^2 - 3\right) dx = \int_0^3 (15 - 2x - x^2) dx$$ Calculate the integral: $$\int_0^3 15 dx = 15x \Big|_0^3 = 45$$ $$\int_0^3 2x dx = x^2 \Big|_0^3 = 9$$ $$\int_0^3 x^2 dx = \frac{x^3}{3} \Big|_0^3 = 9$$ So, $$CS = 45 - 9 - 9 = 27$$ 6. **Calculate producer's surplus:** Producer surplus is the area between the equilibrium price and the supply curve from 0 to equilibrium quantity: $$PS = \int_0^3 (P_e - P_s) \, dx = \int_0^3 (3 - (2x - 3)) dx = \int_0^3 (6 - 2x) dx$$ Calculate the integral: $$\int_0^3 6 dx = 6x \Big|_0^3 = 18$$ $$\int_0^3 2x dx = x^2 \Big|_0^3 = 9$$ So, $$PS = 18 - 9 = 9$$ **Final answers:** Consumer's surplus = $27$ Producer's surplus = $9$