Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

8.2.3 Solving a Quadratic Equation Set Equal to Zero

8.2.3 • Solving a Quadratic Equation Set Equal to Zero

Activity

The expressions $p (200 - 5 p)$ and $- 5 p^{2} + 200 p$ define the same function. The function models the revenue a school would earn from selling raffle tickets at $p$ dollars each.

1.

At what price or prices would the school collect $0 revenue from raffle sales? Explain or show your reasoning.

Compare your answer:

$0 and $40. For example, if the equation $p (200 - 5 p) = 0$ is true, then either $p$ is equal to 0, or $200 - 5 p$ is equal to 0. If $200 - 5 p = 0$ , then $p$ is 40, because $200 - 5 (40)$ is 0.

2.

The school staff noticed that there are two ticket prices that would result in a revenue of $500. What equation could you use to determine what those two prices are?

Compare your answer:

We could write $p (200 - 5 p) = 500$ and $- 5 p^{2} + 200 p = 500$ . Approximate solutions can be found using trial and error or graphing. However, we don’t yet know an exact way to solve either equation.

Are you ready for more?

Extending Your Thinking

1.

Can you find the following prices without graphing?

If the school charges $10, it will collect $1500 in revenue. Find another price that would generate $1500 in revenue.

Compare your answer:

$30

Moving the same distance away from each horizontal intercept, at $0 and $40, results in equal revenue. This means $10 and $30 earn an equal amount of revenue.

2.

If the school charges $28, it will collect $1680 in revenue. Find another price that would generate $1680 in revenue.

Compare your answer:

$12

Moving the same distance away from each horizontal intercept, at $0 and $40, results in equal revenue. This means $12 and $28 earn an equal amount of revenue.

3.

Find the price that would produce the maximum possible revenue. Be prepared to show your reasoning.

Compare your answer:

$20. Moving the same distance away from each horizontal intercept results in equal revenue. The farther away we move from one horizontal intercept in the direction of the other intercept, the greater the revenue. An optimal ticket price of $20, producing $2000 in revenue, is found in the middle point between the two intercepts, at $p = 20$ .

Video: Solving a Quadratic Equation Set Equal to Zero

Watch the following video to learn more about solving a quadratic equation set equal to zero.

8.2.3

Self Check

The expressions $s(60 − 12s)$ and $-12s^2 + 60s$ define the same function. A soap bubble machine can launch many soap bubbles into the air before they begin to pop. The function given models the number of soap bubbles floating in the air in $s$ seconds.

According to the function, how many seconds have passed before all of the bubbles have been made and also popped?

7 seconds
6 seconds
5 seconds
0 seconds

Additional Resources

Solving a Quadratic Equation Set Equal to Zero

The expressions $h (40 - 10 h)$ and $- 10 h^{2} + 40 h$ define the same function. The function models the height of a weather balloon in miles $h$ hours after it is released.

1. What equation can be used to determine the times the weather balloon is on the ground?

Solution

When the weather balloon is on the ground, the expression will equal 0. So, the equation $h (40 - 10 h) = 0$ represents the weather balloon when it is on the ground.

2. How can we solve the equation $h (40 - 10 h) = 0$ ?

Solution

We know that any value multiplied by 0 equals 0. So, if $h$ or $(40 - 10 h)$ equals 0, then the left side of the equation will also equal 0.

The weather balloon is on the ground at 0 hours. Now, let’s find the solution to $40 - 10 h = 0$ .

$40 - 10 h = 0$

$40 = 10 h$

$4 = h$

The weather balloon is also on the ground after 4 hours.

3. A scientist wants to determine the two different times when the weather balloon is 30 miles above Earth. Write an equation to represent these times.

Solution

The times when the weather balloon is 30 miles above Earth are represented by the equation $h (40 - 10 h) = 30$ or the equation $- 10 h^{2} + 40 h = 30$ .

4. If the weather balloon is 30 miles above Earth after 1 hour, find another time that the weather balloon is 30 miles above Earth.

Solution

We know the total trip takes the weather balloon 4 hours. Since the graph representing the flight of the weather balloon is symmetric, if it takes 1 hour to reach 30 miles up, then 1 hour from touching the ground, it will be at 30 miles up again. So, the weather balloon will be 30 miles above Earth after 3 hours.

Try it

Solving a Quadratic Equation Set Equal to Zero

The expressions $p (420 - 60 p)$ and $- 60 p^{2} + 420 p$ define the same function. The function models revenue earned at a baseball game from the sale of baseball magazines sold at $p$ dollars each.

1. At what price or prices would the game organizers collect $0 revenue from magazine sales? Explain or show your reasoning.

Here is how to use your knowledge of quadratic equations to solve the problem:

$0 and $7. If the equation $p (420 - 60 p) = 0$ is true, then either $p$ is equal to 0, or $420 - 60 p$ is equal to 0. If $420 - 60 p = 0$ , then $p$ is 7, because $420 - 60 (7)$ is 0.

2. The organizers determined there are two magazine prices that would result in a revenue of $600. What equation would you use to determine what those two prices are?

Here is how to use your knowledge of quadratic equations to solve the problem:<

We could write $p (420 - 60 p) = 600$ or $- 60 p^{2} + 420 p = 600$ . We can use trial and error or graphing to provide approximate solutions. We do not yet know an exact method to solve either equation.