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Question 1 A)Use the Zero Product Property to find the value of x that makes the following statement true: 2⋅(x−3)=0.(1 point) The value that makes the statement true is x=$$. Question 2 A) Zavier, Boaz, and Sophie are trying to find the values of x that make the following quadratic function equal 0: x(x+2)=0. Each student came up with a different answer, which they explained in the options. Which student’s explanation is correct? Option #1: Zavier says that only x=−2 will make the function equal 0, and he provided the given work: −2⋅(−2+2)=−2⋅(0)=0. Option #2: Boaz says that the values x=0 and x=−2 will both make the function equal 0, and he provided the given work: 0⋅(0+2)=0⋅(2)=0, and −2⋅(−2+2)=−2(0)=0. Option #3: Sophie says that only x=0 will make the function equal 0, and she provided the given work: 0⋅(0+2)=0⋅(2)=0. (1 point) Option #$$ is the correct explanation. Question 3 A) Use the table to answer the question. Option 1 (x−2)(x+3)=7 Option 2 x(x+3)=0 Option 3 (x+3)2−1=0 It is only possible to use the Zero Product Property on one of the options in this table. Which option can the Zero Product Property be used on to find the zeros of the function? (1 point) Option $$ Question 4 A)Solve the factored quadratic equation (x−5)(x+3)=0.(1 point) The smaller solution is x=$$, and the larger solution is x=$$. Question 5 A) To begin a football game, a kicker has to kick off. The football follows the path h(x)=−130x(x−61), where h(x) is the height of the football and x is the horizontal distance it has traveled in yards. Solve the equation to determine how far the ball will have traveled when it hits the ground. 0=−130x(x−61) (1 point) The ball will have traveled $$ yards before hitting the ground.

3 answers

Let's go through each question step by step:

Question 1: To use the Zero Product Property, we set the product equal to zero:

\[ 2 \cdot (x - 3) = 0 \]

For the product to be zero, one of the factors must be zero. Since 2 is not equal to zero, we have:

\[ x - 3 = 0 \] \[ x = 3 \]

The value that makes the statement true is \( x = 3 \).

Question 2: We need to solve \( x(x + 2) = 0 \) using the Zero Product Property. This means:

  1. \( x = 0 \)
  2. \( x + 2 = 0 \) which gives \( x = -2 \)

Now looking at the options:

  • Zavier is incorrect as he only considers \( x = -2 \).
  • Boaz is correct as he mentions both \( x = 0 \) and \( x = -2 \).
  • Sophie is incorrect as she only considers \( x = 0 \).

So, Option #2 is the correct explanation.

Question 3: We must identify which option can be used with the Zero Product Property:

  • Option 1: \( (x - 2)(x + 3) = 7 \) - This is not in the form of products equating to zero.
  • Option 2: \( x(x + 3) = 0 \) - This can be applied to the Zero Product Property.
  • Option 3: \( (x + 3)^2 - 1 = 0 \) - This requires further manipulation to factor it.

Thus, Option 2 can be used with the Zero Product Property.

Question 4: To solve \( (x - 5)(x + 3) = 0 \): Setting each factor to zero gives:

  1. \( x - 5 = 0 \) leads to \( x = 5 \)
  2. \( x + 3 = 0 \) leads to \( x = -3 \)

The smaller solution is \( x = -3 \) and the larger solution is \( x = 5 \).

Question 5: We need to find \( x \) when \( h(x) = 0 \):

\[ 0 = -130x(x - 61) \]

Setting each factor to zero:

  1. \( -130x = 0 \) gives \( x = 0 \)
  2. \( x - 61 = 0 \) gives \( x = 61 \)

Thus the ball will have traveled \( 61 \) yards before hitting the ground, assuming we're looking for positive distances.

Putting it all together:

  1. Question 1: The value that makes the statement true is \( x = 3 \).
  2. Question 2: Option #2 is the correct explanation.
  3. Question 3: Option 2.
  4. Question 4: The smaller solution is \( x = -3 \), and the larger solution is \( x = 5 \).
  5. Question 5: The ball will have traveled \( 61 \) yards before hitting the ground.
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