How to Solve Word Problems Using Quadratic Equations - YouTube - Free Printable

Let’s solve this step by step.

We are told:
- Base price = $300 → gets 1000 members.
- For every $2 increase in price, they lose 5 members.
- We want to find the price that gives maximum revenue.

Revenue = (number of members) × (price per member)

Let’s define:
Let x = number of $2 increases.

Then:
- Price = 300 + 2x
- Members = 1000 - 5x

So Revenue R(x) = (1000 - 5x)(300 + 2x)

Let’s expand this:

R(x) = 1000×300 + 1000×2x - 5x×300 - 5x×2x
= 300,000 + 2000x - 1500x - 10x²
= 300,000 + 500x - 10x²

So R(x) = -10x² + 500x + 300,000

This is a quadratic equation that opens downward (because coefficient of x² is negative), so it has a maximum at its vertex.

Vertex of parabola ax² + bx + c is at x = -b/(2a)

Here, a = -10, b = 500

x = -500 / (2 × -10) = -500 / -20 = 25

So, x = 25 → meaning 25 increases of $2 each.

Therefore, optimal price = 300 + 2×25 = 300 + 50 = $350

Now let’s check if this makes sense with the handwritten work.

The handwritten work shows:

Revenue = (1000 - 5x)(300 + 2x)

Then factored as:
5(200 - x) × 2(150 + x) → which is correct because:
1000 - 5x = 5(200 - x)
300 + 2x = 2(150 + x)

So R(x) = 5×2 × (200 - x)(150 + x) = 10(200 - x)(150 + x)

Then rewritten as:
10(-x + 200)(x + 150) → same thing.

Then written as:
-10(x - 200)(x + 150) → also correct, since factoring out -1 from first term:
(-x + 200) = -(x - 200), so overall becomes -10(x - 200)(x + 150)

But note: when we expand -10(x - 200)(x + 150), we get:

First multiply (x - 200)(x + 150) = x² -50x -30000
Then multiply by -10: -10x² + 500x + 300,000 → matches our earlier expansion.

So both methods agree.

Maximum occurs at vertex. Since it's factored as -10(x - 200)(x + 150), the roots are x = 200 and x = -150. The vertex is halfway between them:

x = (200 + (-150)) / 2 = 50/2 = 25 → again, x = 25.

So price = 300 + 2×25 = $350

Maximum revenue = (1000 - 5×25)(300 + 2×25) = (1000 - 125)(350) = 875 × 350

Calculate that:
875 × 350 = 875 × (300 + 50) = 875×300 + 875×50
= 262,500 + 43,750 = 306,250

So maximum revenue is $306,250 at price $350.

Final Answer:
$350