If the supply function for a commodity is given by P = q^2 + 100 and function is given by 
P = -20q + 2500 find the point of market equilibrium.

To find the point of market equilibrium, we need to find the values of \( q \) and \( P \) where the supply function equals the demand function.

Given:
- **Supply function**: \( P = q^2 + 100 \)
- **Demand function**: \( P = -20q + 2500 \)

At equilibrium, the quantity supplied equals the quantity demanded, so we set the two equations equal to each other:

\[
q^2 + 100 = -20q + 2500
\]

Now, let's solve this equation for \( q \).

1. Rearrange the equation:
\[
q^2 + 20q + 100 - 2500 = 0
\]
\[
q^2 + 20q - 2400 = 0
\]

2. Solve the quadratic equation \( q^2 + 20q - 2400 = 0 \) using the quadratic formula:
\[
q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 20 \), and \( c = -2400 \).

\[
q = \frac{-20 \pm \sqrt{20^2 - 4(1)(-2400)}}{2(1)}
\]
\[
q = \frac{-20 \pm \sqrt{400 + 9600}}{2}
\]
\[
q = \frac{-20 \pm \sqrt{10000}}{2}
\]
\[
q = \frac{-20 \pm 100}{2}
\]

So, we have two solutions for \( q \):
\[
q = \frac{-20 + 100}{2} = \frac{80}{2} = 40
\]
\[
q = \frac{-20 - 100}{2} = \frac{-120}{2} = -60
\]

Since quantity \( q \) cannot be negative in this context, the only valid solution is \( q = 40 \).

3. Now, substitute \( q = 40 \) into either the supply or demand function to find the equilibrium price \( P \).

Using the supply function:
\[
P = q^2 + 100 = 40^2 + 100 = 1600 + 100 = 1700
\]

Thus, the market equilibrium occurs when \( q = 40 \) and \( P = 1700 \).

### Conclusion:
The point of market equilibrium is:
\[
(q, P) = (40, 1700)
\]

Would you like further details or explanations?

### Related Questions:
1. What happens to the equilibrium point if the demand function changes to \( P = -15q + 2400 \)?
2. How would taxes affect the equilibrium in this market?
3. What is the elasticity of demand at the equilibrium point?
4. How would a price ceiling below \( 1700 \) affect the market?
5. How can we find the consumer and producer surplus at the equilibrium?

### Tip:
Always ensure to distinguish between valid and invalid roots, especially in real-world applications like quantity in economics.

If the supply function for a commodity is given by P = q^2 + 100 and the demand function is given by P = -20q + 2500, find the point of market equilibrium.

Discover how to find the point of market equilibrium by solving a quadratic equation. Given supply and demand functions, learn to solve for equilibrium quantity and price using algebraic methods. This guide is suitable for high school students (Grades 9-12) and covers key concepts like the quadratic equation formula and equilibrium theory in economics.

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