Solving Quadratic Equations with the Zero Product Property and D

Solving Quadratic Equations with the Zero Product Property and Determining the Number of Solutions

• Let’s find solutions to equations that contain products that equal zero.

• Let’s use graphs to investigate quadratic equations that have two solutions, one solution, or no solutions.

•  can explain the meaning of the “zero product property.”

• I can find solutions to quadratic equations when one side is a product of factors and the other side is zero.

• I can explain why dividing by a variable to solve a quadratic equation is not a good strategy.

• I know that quadratic equations can have no solutions and can explain why there are none.

zero product property

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.

Dropdown

We have seen quadratic functions modeling the height of a projectile as a function of time.

Here are two ways to define the same function that approximates the height of a projectile in meters, t seconds after launch:

A)​ h(t) = -5t² + 27t + 18 B) h(t) = (-5t – 3)(t – 6)

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Which way of defining the function allows us to u​se the zero product property to find out when the height of the object is 0 meters?​ ​ ​ ​

Select answer

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Fill in the Blank

Here are two ways to define the same function that approximates the height of a projectile in meters, t seconds after launch:

​ h(t) = -5t² + 27t + 18 h(t) = (-5t – 3)(t – 6)

Without graphing, determine at what time the height of the object is 0 meters.

Multiple Choice

All solutions to (x + 5)(2x – 3) = 0 are. 

• x = 5 and x = 2/3

• x = -5 and x = -3/2

• x = -5 and x = -3/2

• x=-5 and x = 3/2

Solving Quadratic Equations with the Zero Product Property

• The zero product property helps us find the solutions to (x – 3)(x + 4) = 0, by showing us that either x-3=0, or x+4 = 0.?

• The zero product property only works when the product of the factors is zero. When the product is any other number, we can’t conclude that each factor is that number; therefore why the solutions to (x – 3)(x + 4) = 8 will not be 3 and -4?

• Although he expression x² – x – 12 is equivalent to (x + 3)(x – 4). We cannot apply the zero product property to solve x² – x – 12 = 0 because the expression is not a product of factors.

• You cannot solve the quadratic x² – x – 12 = 0 by performing the same operation to each side of the equation because doing so doesn’t help us isolate the variable.?

Dropdown

3 is the only solution to x² – 9 = 0. ​ ​

Select answer

Dropdown

x(x – 7) = 0 has two solutions​​

Select answer

Dropdown

5 and 7 are the solutions to (x – 5)(x + 7) = 12 ​ ​

Select answer

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Open Ended

If we were solving three equations by graphing.

(x – 5)(x – 3) = 0 (x – 5)(x – 3) = -1 (x – 5)(x – 3) = -4

To solve the first equation, (x – 5)(x – 3) = 0, we need to graph y = (x – 5)(x – 3)

What are the solutions?

Open Ended

If we were solving three equations by graphing.

(x – 5)(x – 3) = 0 (x – 5)(x – 3) = -1 (x – 5)(x – 3) = -4

To solve the second equation, we need to rewrite it as (x – 5)(x – 3) + 1 = 0. We then graph y = (x – 5)(x – 3) + 1.

What are the solutions?

Open Ended

Solve the third equation (x – 5)(x – 3) = -4 using the previous strategies to find the solutions, what are they?

Open Ended

Using the previous strategies graph each equation to show the solution. Place a copy of your graph as your answer.

x² = 121

x² – 31 = 5

(x – 4)(x – 4) = 0

(x + 3)(x – 1) = 5

(x + 1)² = -4

Dropdown

Consider (x – 5)(x + 1) = 7. Priya reasons that if this is true, then either    x – 5 = 7 or x + 1 = 7. So, the solutions to the original equation are 12 and 6.

Do you agree or disagree?​

Select answer

 (x – 5)(x + 1) = 7​

Disagree. Priya solved the equation using the reasoning we would use with the zero product property, but the zero product property only works if the product of two factors is 0. We can tell that 12 isn’t a solution because is 91, not 7.

Open Ended

We cannot we use the zero product property to solve x(x – 250) = 100 because the expression on the left does not equal 0.

We can solve x(x – 250) = 100 by graphing

What does the graph tell us about how many solutions and what are they?

(Place a copy of your graph with your answer.)

•  can explain the meaning of the “zero product property.”

• I can find solutions to quadratic equations when one side is a product of factors and the other side is zero.

• I can explain why dividing by a variable to solve a quadratic equation is not a good strategy.

• I know that quadratic equations can have no solutions and can explain why there are none.