Given the demend equation \( \begin{array}{rl}P & P=-q_{d}^{2}-4 q_{d}+68 \text { and } \\ & P=-q_{s}^{2}-2 q_{s}+12 \text { Determine the } \\ & P_{s} \text { and } s\end{array} \)

To determine the equilibrium price (\( P_s \)) and equilibrium quantity (\( s \)) in the market, we'll set the demand and supply equations equal to each other and solve for the quantity. Here's a step-by-step breakdown: ### Given Equations: 1. **Demand Equation:** \[ P = -q_d^2 - 4q_d + 68 \] Where: - \( P \) = Price - \( q_d \) = Quantity Demanded 2. **Supply Equation:** \[ P = -q_s^2 - 2q_s + 12 \] Where: - \( q_s \) = Quantity Supplied ### Step 1: Set Demand Equal to Supply for Equilibrium At equilibrium, quantity demanded equals quantity supplied (\( q_d = q_s = q \)) and prices are equal. Therefore: \[ -q^2 - 4q + 68 = -q^2 - 2q + 12 \] ### Step 2: Simplify the Equation \[ -4q + 68 = -2q + 12 \] Subtract \( -2q \) from both sides: \[ -2q + 68 = 12 \] Subtract 68 from both sides: \[ -2q = -56 \] Divide both sides by -2: \[ q = 28 \] ### Step 3: Determine the Equilibrium Price (\( P_s \)) Plug the equilibrium quantity back into either the demand or supply equation. Using the supply equation: \[ P_s = -q_s^2 - 2q_s + 12 \] \[ P_s = -(28)^2 - 2(28) + 12 \] \[ P_s = -784 - 56 + 12 \] \[ P_s = -828 \] ### Interpretation: - **Equilibrium Quantity (\( s \)):** 28 units - **Equilibrium Price (\( P_s \)):** \(-828\) ### Observations: 1. **Negative Price:** The calculated equilibrium price is negative, which doesn't make economic sense in a real-world context. This suggests that there might be an issue with the original equations provided. Negative prices are generally not feasible in standard market scenarios. 2. **Possible Reasons for Negative Price:** - **Typographical Errors:** There might be a mistake in the sign conventions of the provided equations. - **Model Assumptions:** If the equations are part of a theoretical model with specific assumptions, ensure that they align with real-world behavior. - **Non-viable Market Conditions:** The structure of the demand and supply might indicate an unsustainable market condition under the given equations. ### Recommendation: Review the original demand and supply equations for potential errors or reconsider the model's assumptions to ensure that they produce economically meaningful results.