5. Find the equilibrium values for the following market equations
D= 50 - 10 P² and S =- 75 + 50 P.


To find the equilibrium values for the given market equations, we need to solve for the price (P) where the quantity demanded (D) equals the quantity supplied (S). 

The equations are:

Demand equation: \( D = 50 - 10P^2 \)
Supply equation: \( S = -75 + 50P \)

At equilibrium, \( D = S \). Therefore, we set the two equations equal to each other:

\[
50 - 10P^2 = -75 + 50P
\]

Now, let's solve for \( P \):

### Step 1: Rearrange the equation
\[
50 - 10P^2 = -75 + 50P
\]
Move all terms to one side:
\[
10P^2 + 50P - 125 = 0
\]

### Step 2: Simplify the equation
Divide the entire equation by 5 to simplify:
\[
2P^2 + 10P - 25 = 0
\]

### Step 3: Solve the quadratic equation using the quadratic formula
The quadratic formula is given by:
\[
P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2P^2 + 10P - 25 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 10 \)
- \( c = -25 \)

Now, substitute these values into the quadratic formula:
\[
P = \frac{-10 \pm \sqrt{10^2 - 4(2)(-25)}}{2(2)}
\]
\[
P = \frac{-10 \pm \sqrt{100 + 200}}{4}
\]
\[
P = \frac{-10 \pm \sqrt{300}}{4}
\]
\[
P = \frac{-10 \pm 10\sqrt{3}}{4}
\]
\[
P = \frac{-5 \pm 5\sqrt{3}}{2}
\]

### Step 4: Calculate the two possible values of \( P \)
The two possible values for \( P \) are:
\[
P_1 = \frac{-5 + 5\sqrt{3}}{2}
\]
\[
P_2 = \frac{-5 - 5\sqrt{3}}{2}
\]

Since price cannot be negative in most economic contexts, we discard \( P_2 \) and keep \( P_1 \). 

Thus, the equilibrium price is:
\[
P = \frac{-5 + 5\sqrt{3}}{2} \approx 1.33
\]

### Step 5: Find the equilibrium quantity
Now substitute this value of \( P \) back into either the demand or supply equation to find the equilibrium quantity. Let’s use the supply equation:

\[
S = -75 + 50P
\]
\[
S = -75 + 50(1.33)
\]
\[
S = -75 + 66.5 = -8.5
\]

Thus, the equilibrium price is approximately \( P = 1.33 \) and the equilibrium quantity is approximately \( Q = -8.5 \).

Let me know if you'd like more details or have any questions!

Here are some related questions:
1. How does a change in the demand curve affect equilibrium price and quantity?
2. What happens if the supply curve shifts to the right?
3. How do you interpret negative equilibrium quantities in economic theory?
4. Can the quadratic formula always be used for solving nonlinear demand equations?
5. What are the assumptions behind equilibrium in supply and demand models?

**Tip**: When solving quadratic equations in economics, always consider the feasibility of the result (e.g., whether negative values make sense in context).

5. Find the equilibrium values for the following market equations D = 50 - 10P² and S = -75 + 50P.

This tutorial shows how to solve for market equilibrium using the quadratic demand equation D = 50 - 10P² and supply equation S = -75 + 50P. Learn step-by-step how to apply the quadratic formula to find equilibrium price and quantity. Suitable for high school economics and math students (Grades 10-12).

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