Use the table to answer the question.

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

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

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 .

Yes, the work is accurate.

Answer 1

Oliver's work contains an error in the second row where he wrote "x−3=0" instead of "x+3=0." The correct steps using the Zero Product Property for the equation \((x+3)(x+4)=0\) should involve setting each factor equal to zero:

1. \(x+3=0\)
2. \(x+4=0\)

Thus, the correct values of \(x\) would be \(x=-3\) and \(x=-4\), not \(x=3\) and \(x=-4\).

The most accurate response to the question is:

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

Answer 2

Based on the Zero Product Property, which of the following statements must be true about the quadratic equation (2x−3)(x+2)=0
?(1 point)
Responses

No values of x will make the function equal 0.
No values of x will make the function equal 0.

Neither of the factors, (2x−3)
nor ​(x+2)
, must equal 0 for the entire function to equal 0.
Neither of the factors, left parenthesis 2 x minus 3 right parenthesis nor left parenthesis x plus 2 right parenthesis , must equal 0 for the entire function to equal 0.

At least one of the factors, (2x−3)
or ​(x+2)
, must equal 0 for the entire function to equal 0.
At least one of the factors, left parenthesis 2 x minus 3 right parenthesis or left parenthesis x plus 2 right parenthesis , must equal 0 for the entire function to equal 0.

Both factors, (2x−3)
and ​(x+2)
, must equal 0 for the entire function to equal 0.

Answer 3

Based on the Zero Product Property, when you have a product of two factors equal to zero (in this case, \((2x−3)(x+2)=0\)), at least one of the factors must equal zero. This means:

• If one factor equals zero, the product can equal zero.
• Therefore, you do not need both factors to equal zero simultaneously; only one of them needs to be zero for the entire equation to hold true.

So, the correct statement is:

At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.

Answer 4

The path of a soccer ball can be modeled by the equation h(x)=−12x(x−20)
, where h(x) is the height of the ball, and x is the horizontal distance the ball has traveled. To determine how far the ball has traveled when it hits the ground, Leon set the height equal to 0 and obtained the equation: 0=−12x(x−20)
. How can he use the Zero Product Property to find the horizontal distance the ball has traveled when the height of the soccer ball equals 0?(1 point)
Responses

Solve the two equations: −12x=0
and x−20=0
.
Solve the two equations: negative Start Fraction 1 over 2 End Fraction x equals 0 and x minus 20 equals 0 .

Plug in 0 for all the x terms in the function.
Plug in 0 for all the x terms in the function.

He cannot use the Zero Product Property because the height will never be 0.
He cannot use the Zero Product Property because the height will never be 0.

Cancel out the x and then multiply −12
and −20
.

Answer 5

To use the Zero Product Property to find the horizontal distance the soccer ball has traveled when its height is 0, Leon needs to set each factor of the equation \(0 = -12x(x - 20)\) to zero.

The equation can be rewritten as:

\[ -12x = 0 \quad \text{or} \quad x - 20 = 0 \]

This leads to the two equations he needs to solve:

1. \(-12x = 0\) which simplifies to \(x = 0\)
2. \(x - 20 = 0\) which simplifies to \(x = 20\)

This means the correct response is:

Solve the two equations: −12x=0 and x−20=0.

Leon will find that the ball hits the ground when it has traveled either 0 units (the starting point) or 20 units (the point where it hits the ground).

Answer 6

Solve the factored quadratic equation 0=−13x(x−6)
.(1 point)
Responses

The solution set is x={−6,3}
.
The solution set is x equals left brace neagtive 6 comma 3 right brace .

The solution set is x={0,−6}
.
The solution set is x equals left brace neagtive 0 comma negative 6 right brace .

The solution set is x={0,6}
.
The solution set is x equals left brace 0 comma 6 right brace .

The solution set is x={13,6}
.
The solution set is x equals left brace Start Fraction 1 over 3 End Fraction comma 6 right brace .

Answer 7

To solve the factored quadratic equation \(0 = -13x(x - 6)\), we can use the Zero Product Property. This means we set each factor equal to zero.

1. From the first factor: \(-13x = 0\) \(\Rightarrow x = 0\)

2. From the second factor: \(x - 6 = 0\) \(\Rightarrow x = 6\)

Therefore, the solutions are \(x = 0\) and \(x = 6\).

So, the correct solution set is:

The solution set is x={0,6}.

Answer 8

A golfer hits a golf ball toward the hole 55 yards away. The ball follows a parabolic path given by the function h(x)=−125x(x−45)
, where h(x)
is the height of the golf ball and x is the horizontal distance it has traveled. Solve the equation to determine how far the ball will have traveled when it hits the ground. How far will the ball need to roll to make it to the hole?

0=−125x(x−45)

(1 point)
Responses

The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.
The ball will hit the ground after it has traveled 30 yards. It will need to roll an additional 25 yards to reach the hole.

The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.
The ball will hit the ground after it has traveled 25 yards. It will need to roll an additional 30 yards to reach the hole.

The ball will hit the ground after it has traveled 10 yards. It will need to roll an additional 45 yards to reach the hole.
The ball will hit the ground after it has traveled 10 yards. It will need to roll an additional 45 yards to reach the hole.

The ball will hit the ground after it has traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.

Answer 9

To determine how far the golf ball has traveled when it hits the ground, we start with the equation:

\[ 0 = -125x(x - 45) \]

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

1. From the first factor: \(-125x = 0\) \(\Rightarrow x = 0\)

2. From the second factor: \(x - 45 = 0\) \(\Rightarrow x = 45\)

The ball hits the ground at two points:

• \(x = 0\) (when it was hit)
• \(x = 45\) (where it hits the ground).

Now, since the hole is 55 yards away, we can determine how much further the ball will need to roll after it hits the ground at 45 yards:

\[ \text{Distance to hole} = 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]

Based on this analysis, the correct response is:

The ball will hit the ground after it has traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.