Question: The supply and demand for a particular market is given by the equations
$P_s = 96 + 2Q + Q^2$ and $P_d = 25 + 10Q - 2Q^2$.
1.2.1 Sketch the graph of each function over the interval $Q = 0$ to $Q = 5$.
1.2.2 Find the equilibrium price and quantity graphically and algebraically
1. **State the problem:**
We have supply and demand functions:
$$P_s = 96 + 2Q + Q^2$$
$$P_d = 25 + 10Q - 2Q^2$$
We want to:
- Sketch both functions for $Q$ from 0 to 5.
- Find the equilibrium price and quantity where supply equals demand.
2. **Sketching the graphs:**
- For $Q$ in $[0,5]$, calculate $P_s$ and $P_d$ values.
- Supply $P_s$ is a quadratic increasing function.
- Demand $P_d$ is a quadratic with a negative $Q^2$ term, so it is concave down.
3. **Finding equilibrium algebraically:**
At equilibrium, supply price equals demand price:
$$P_s = P_d$$
Substitute:
$$96 + 2Q + Q^2 = 25 + 10Q - 2Q^2$$
4. **Rearrange the equation:**
$$96 + 2Q + Q^2 - 25 - 10Q + 2Q^2 = 0$$
$$ (Q^2 + 2Q^2) + (2Q - 10Q) + (96 - 25) = 0$$
$$3Q^2 - 8Q + 71 = 0$$
5. **Solve the quadratic equation:**
Use the quadratic formula:
$$Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
where $a=3$, $b=-8$, $c=71$.
Calculate discriminant:
$$\Delta = (-8)^2 - 4 \times 3 \times 71 = 64 - 852 = -788$$
6. **Interpretation:**
Since $\Delta < 0$, there are no real solutions.
This means algebraically, no equilibrium price and quantity exist where supply equals demand in real numbers.
7. **Graphical interpretation:**
- Plotting the curves from $Q=0$ to $Q=5$ shows supply curve is always above demand curve.
- No intersection point means no equilibrium in this interval.
**Final answer:**
- No equilibrium price and quantity exist where $P_s = P_d$ for real $Q$.
- Graphs do not intersect on $[0,5]$.