1. **Problem Statement:** Find the market equilibrium price and quantity given the demand equation $p - 3q = 22$ and the supply equation $q^2 + 2p + 4q = 100$, where $p$ is price and $q$ is quantity. 2. **Step 1: Express $p$ from the demand equation.** From $p - 3q = 22$, we get: $$p = 3q + 22$$ 3. **Step 2: Substitute $p$ into the supply equation.** Supply equation: $q^2 + 2p + 4q = 100$ Substitute $p = 3q + 22$: $$q^2 + 2(3q + 22) + 4q = 100$$ Simplify: $$q^2 + 6q + 44 + 4q = 100$$ $$q^2 + 10q + 44 = 100$$ 4. **Step 3: Rearrange to form a quadratic equation.** $$q^2 + 10q + 44 - 100 = 0$$ $$q^2 + 10q - 56 = 0$$ 5. **Step 4: Solve the quadratic equation using the quadratic formula:** $$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=10$, $c=-56$. Calculate discriminant: $$\Delta = 10^2 - 4(1)(-56) = 100 + 224 = 324$$ Square root: $$\sqrt{324} = 18$$ Calculate roots: $$q = \frac{-10 \pm 18}{2}$$ Two solutions: $$q_1 = \frac{-10 + 18}{2} = \frac{8}{2} = 4$$ $$q_2 = \frac{-10 - 18}{2} = \frac{-28}{2} = -14$$ Since quantity cannot be negative, $q = 4$. 6. **Step 5: Find equilibrium price $p$ using $p = 3q + 22$.** $$p = 3(4) + 22 = 12 + 22 = 34$$ 7. **Step 6: Calculate total revenue (TR).** Total revenue is price times quantity: $$TR = p \times q = 34 \times 4 = 136$$ **Final answers:** - Market equilibrium quantity: $4$ - Market equilibrium price: $34$ - Total revenue: $136$