Question
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
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.
Answers
Answer
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
Neither of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Neither 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.
No values of x will make the function equal 0.
No values of x will make the function equal 0.
Both factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Both 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.
At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Responses
Neither of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Neither 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.
No values of x will make the function equal 0.
No values of x will make the function equal 0.
Both factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Both 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.
At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.
Answer
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
Plug in 0 for all the x terms in the function.
Plug in 0 for all the x terms in the function.
Cancel out the x and then multiply −12 and −20.
Cancel out the x and then multiply negative Start Fraction 1 over 2 End Fraction and negative 20 .
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.
Solve the two equations: −12x=0 and x−20=0.
Responses
Plug in 0 for all the x terms in the function.
Plug in 0 for all the x terms in the function.
Cancel out the x and then multiply −12 and −20.
Cancel out the x and then multiply negative Start Fraction 1 over 2 End Fraction and negative 20 .
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.
Solve the two equations: −12x=0 and x−20=0.
Answer
Solve the factored quadratic equation 0=−13x(x−6).(1 point)
Responses
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 .
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}.
Responses
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 .
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}.
Answer
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)
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 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 45 yards. It will need to roll an additional 10 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.
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.
0=−125x(x−45)
(1 point)
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 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 45 yards. It will need to roll an additional 10 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.
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.
Answered by
GPT-4o mini
The correct response is:
**No, x−3=0 in the second row should be x+3=0.**
The Zero Product Property states that if the product of two expressions equals zero, then at least one of the expressions must be equal to zero. In this case, for the equation \((x+3)(x+4) = 0\), the correct factors should give us \(x + 3 = 0\) and \(x + 4 = 0\) to find the solutions for \(x\). Therefore, the second row where it shows \(x−3=0\) is incorrect.
**No, x−3=0 in the second row should be x+3=0.**
The Zero Product Property states that if the product of two expressions equals zero, then at least one of the expressions must be equal to zero. In this case, for the equation \((x+3)(x+4) = 0\), the correct factors should give us \(x + 3 = 0\) and \(x + 4 = 0\) to find the solutions for \(x\). Therefore, the second row where it shows \(x−3=0\) is incorrect.
Answered by
GPT-4o mini
The correct response is:
**At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.**
According to the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be equal to zero. Therefore, either \(2x - 3 = 0\) or \(x + 2 = 0\) must hold true for the expression \((2x−3)(x+2) = 0\) to be true.
**At least one of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.**
According to the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be equal to zero. Therefore, either \(2x - 3 = 0\) or \(x + 2 = 0\) must hold true for the expression \((2x−3)(x+2) = 0\) to be true.
Answered by
GPT-4o mini
The correct response is:
**Solve the two equations: −12x=0 and x−20=0.**
To use the Zero Product Property, Leon can set each factor in the equation \(0 = -12x(x - 20)\) equal to zero. This results in the two equations:
1. \(-12x = 0\)
2. \(x - 20 = 0\)
Solving these equations will provide the values for \(x\) that indicate how far the ball has traveled when it hits the ground (when the height is 0).
**Solve the two equations: −12x=0 and x−20=0.**
To use the Zero Product Property, Leon can set each factor in the equation \(0 = -12x(x - 20)\) equal to zero. This results in the two equations:
1. \(-12x = 0\)
2. \(x - 20 = 0\)
Solving these equations will provide the values for \(x\) that indicate how far the ball has traveled when it hits the ground (when the height is 0).
Answered by
GPT-4o mini
To solve the equation \(0 = -13x(x - 6)\), we use the Zero Product Property. We set each factor equal to zero:
1. \(-13x = 0\)
2. \(x - 6 = 0\)
Solving these equations:
1. From \(-13x = 0\), we get \(x = 0\).
2. From \(x - 6 = 0\), we get \(x = 6\).
Thus, the solution set is \(x = \{0, 6\}\).
The correct response is:
**The solution set is x={0,6}.**
1. \(-13x = 0\)
2. \(x - 6 = 0\)
Solving these equations:
1. From \(-13x = 0\), we get \(x = 0\).
2. From \(x - 6 = 0\), we get \(x = 6\).
Thus, the solution set is \(x = \{0, 6\}\).
The correct response is:
**The solution set is x={0,6}.**
Answered by
GPT-4o mini
To solve the equation \(0 = -125x(x - 45)\) using the Zero Product Property, we set each factor to zero:
1. \(-125x = 0\)
2. \(x - 45 = 0\)
Solving these equations gives us:
1. From \(-125x = 0\), we find \(x = 0\).
2. From \(x - 45 = 0\), we find \(x = 45\).
Thus, the ball will hit the ground after it has traveled **45 yards**.
Now, to determine how far the ball needs to roll to reach the hole, which is 55 yards away, we calculate:
\[
55 \text{ yards (distance to the hole)} - 45 \text{ yards (distance traveled)} = 10 \text{ yards}
\]
Thus, the ball will need to roll an additional **10 yards** to reach the hole.
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.**
1. \(-125x = 0\)
2. \(x - 45 = 0\)
Solving these equations gives us:
1. From \(-125x = 0\), we find \(x = 0\).
2. From \(x - 45 = 0\), we find \(x = 45\).
Thus, the ball will hit the ground after it has traveled **45 yards**.
Now, to determine how far the ball needs to roll to reach the hole, which is 55 yards away, we calculate:
\[
55 \text{ yards (distance to the hole)} - 45 \text{ yards (distance traveled)} = 10 \text{ yards}
\]
Thus, the ball will need to roll an additional **10 yards** to reach the hole.
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.**
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