1. **State the problem:** Find the equilibrium price and quantity where demand equals supply. 2. **Given functions:** Demand price $p=8-D^2$ and supply price $p=D^2+6D$. 3. **Equilibrium condition:** At equilibrium, demand price equals supply price: $$8-D^2 = D^2 + 6D$$ 4. **Solve for $D$:** $$8 - D^2 = D^2 + 6D$$ $$8 = 2D^2 + 6D$$ $$2D^2 + 6D - 8 = 0$$ Divide entire equation by 2: $$D^2 + 3D - 4 = 0$$ 5. **Factor the quadratic:** $$(D + 4)(D - 1) = 0$$ 6. **Find roots:** $$D = -4 \quad \text{or} \quad D = 1$$ 7. **Interpret roots:** Quantity $D$ cannot be negative, so $D=1$ is the equilibrium quantity. 8. **Find equilibrium price:** Substitute $D=1$ into either function, e.g., demand: $$p = 8 - (1)^2 = 8 - 1 = 7$$ **Final answer:** Equilibrium quantity is $D=1$ and equilibrium price is $p=7$.