Use the Zero Product Property to find the value of x that makes the following statement true: 2⋅(x−3)=0 .(1 point)

Question

The value that makes the statement true is x=
.

Answer 1

To use the Zero Product Property, we start with the equation:

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

According to the Zero Product Property, if the product of two factors is zero, at least one of the factors must be equal to zero. In this case, we have:

1. The first factor, which is 2, cannot be equal to zero since it is a nonzero constant.
2. Therefore, we examine the second factor:

\[ x - 3 = 0 \]

To solve for \( x \), we add 3 to both sides:

\[ x = 3 \]

Thus, the value that makes the statement true is:

\[ x = 3 \]

Answer 2

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)

Answer 3

To find the values of \( x \) that make the quadratic function \( x(x + 2) = 0 \) true, we can apply the Zero Product Property. This property states that if a product of two factors equal zero, at least one of the factors must be equal to zero.

Given the equation:

\[ x(x + 2) = 0 \]

We can set each factor equal to zero:

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

Thus, the solutions to the equation are \( x = 0 \) and \( x = -2 \).

Now, reviewing the students' explanations:

• Zavier's Explanation: Claims only \( x = -2 \) makes the function equal 0, but he is only considering one factor. His conclusion is incorrect because he omits \( x = 0 \).

• Boaz's Explanation: Correctly identifies that both \( x = 0 \) and \( x = -2 \) will make the function equal 0. He provides accurate calculations for both values.

• Sophie's Explanation: Claims that only \( x = 0 \) makes the function equal 0, which is incorrect because she also overlooks \( x = -2 \).

Based on the analysis, Boaz's explanation is the correct one as he identifies both values \( x = 0 \) and \( x = -2 \) as solutions to the equation \( x(x + 2) = 0 \).

Answer 4

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)

Answer 5

The Zero Product Property can only be applied when an equation is in the form of a product of factors equal to zero. Let's evaluate each option:

1. Option 1: \((x−2)(x+3)=7\)

   This is not set equal to zero. We cannot apply the Zero Product Property here because the equation needs to be equal to zero in order to use that property.

2. Option 2: \(x(x+3)=0\)

   This is already set up as a product of factors equal to zero. We can indeed apply the Zero Product Property here, which tells us that either \(x = 0\) or \(x + 3 = 0\). This option is valid for finding the zeros of the function.

3. Option 3: \((x+3)^2−1=0\)

   While this equation could potentially be solved, it does not fit the form of a product equal to zero. To use the Zero Product Property, we would first need to factor this expression accordingly.

Given this analysis, Option 2: \(x(x+3)=0\) is the only option where the Zero Product Property can be applied to find the zeros of the function.

Answer 6

Solve the factored quadratic equation (x−5)(x+3)=0
.(1 point)
The smaller solution is x=
, and the larger solution is x=
.

Answer 7

To solve the factored quadratic equation \((x - 5)(x + 3) = 0\), we can use the Zero Product Property, which states that if the product of two factors equals zero, at least one of the factors must be equal to zero.

We set each factor equal to zero:

1. \(x - 5 = 0\)

   Solving for \(x\) gives: \[ x = 5 \]

2. \(x + 3 = 0\)

   Solving for \(x\) gives: \[ x = -3 \]

Now we have two solutions: \(x = 5\) and \(x = -3\).

To identify which is the smaller and the larger solution:

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

Thus, the final answers are:

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

Answer 8

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=−1/30x(x−61)


(1 point)
The ball will have traveled
yards before hitting the ground.

Answer 9

To determine how far the ball will have traveled when it hits the ground, we need to solve the equation \( h(x) = 0 \). Given the height function:

\[ h(x) = -\frac{1}{30}x(x - 61) \]

Setting this equal to zero:

\[ 0 = -\frac{1}{30}x(x - 61) \]

To solve this equation, we can use the Zero Product Property, which tells us that if a product is equal to zero, at least one of the factors must be equal to zero. To simplify, we can neglect the \(-\frac{1}{30}\) since it does not affect the solutions:

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

Setting each factor equal to zero:

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

Now we have two solutions: \(x = 0\) and \(x = 61\).

Since \(x = 0\) corresponds to the moment of the kick-off and \(x = 61\) represents the distance traveled when the ball hits the ground, the ball will have traveled 61 yards before hitting the ground.

Thus, the final answer is:

The ball will have traveled 61 yards before hitting the ground.

Answer 10

Use the table to answer the question.

(x+3)(x+4)=0
x−3=0 x+4=0
x=3 x=−4 While using the Zero Product Property to find the values of x that make the quadratic equation (x+3)(x+4) equals 0, Oliver completed the work provided in the table. Is Oliver’s work accurate?(1 point)
Responses

Yes, the work is accurate.
Yes, the work is accurate.

No, he should have a third column where he should set x=0.
No, he should have a third column where he should set x equals 0 .

No, x+4=0 in the second row should be x−4=0.
No, x plus 4 equals 0 in the second row should be x minus 4 equals 0 .

No, x−3=0 in the second row should be x+3=0.

Answer 11

Let's analyze the work in the table and determine if Oliver's work is accurate.

The equation given is:

\[ (x + 3)(x + 4) = 0 \]

To apply the Zero Product Property, we set each factor equal to zero:

1. From \(x + 3 = 0\), solving gives \(x = -3\).
2. From \(x + 4 = 0\), solving gives \(x = -4\).

Now let's check the provided rows:

• The row "x - 3 = 0" is incorrect. It should be "x + 3 = 0" since we derived this from the factor \(x + 3\).
• The row "x + 4 = 0" is correct; however, the solution to \(x + 4 = 0\) gives \(x = -4\), which is correct.

Now, let's evaluate the responses:

1. Yes, the work is accurate.

   • This is incorrect because of the mistake in the first row.

2. No, he should have a third column where he should set x=0.

   • This option is misleading. Setting \(x = 0\) does not apply since we find zeros of the factors, not just setting them to zero.

3. No, x + 4 = 0 in the second row should be x - 4 = 0.

   • This is incorrect. The separation of factors is correct; the factor \(x + 4\) does equal \(0\).

4. No, x - 3 = 0 in the second row should be x + 3 = 0.

   • This is correct because Oliver incorrectly wrote \(x - 3 = 0\) instead of \(x + 3 = 0\).

Based on this analysis, the best answer is:

No, x - 3 = 0 in the second row should be x + 3 = 0.

Oliver's work is not accurate due to the incorrect equation present in the table.