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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.

7 answers

To determine how far the ball will have traveled when it hits the ground, we need to solve the equation:

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

This equation is already factored. We can set each factor equal to zero:

  1. \( -130 = 0 \) (This does not provide any solutions since it's a constant.)
  2. \( x = 0 \)
  3. \( x - 61 = 0 \) which simplifies to \( x = 61 \)

Thus, the two solutions are:

  1. \( x = 0 \): This corresponds to the point where the ball is kicked off the ground.
  2. \( x = 61 \): This is the distance the ball has traveled when it hits the ground.

Therefore, the ball will have traveled 61 yards before hitting the ground.

..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 . Yes, the work is accurate. Yes, the work is accurate. No, x+4=0 in the second row should be x−4=0.

To determine if Oliver's work is accurate, let's apply the Zero Product Property to the given quadratic equation:

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

According to the Zero Product Property, if the product of two factors equals zero, at least one of the factors must equal zero. Therefore, we set each factor equal to zero:

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

Now let's solve these equations:

  1. For \( x + 3 = 0 \): \[ x = -3 \]

  2. For \( x + 4 = 0 \): \[ x = -4 \]

So the solutions to the equation \((x + 3)(x + 4) = 0\) are \( x = -3 \) and \( x = -4 \).

Now, evaluating the given responses:

  • No, x−3=0 in the second row should be x+3=0. (This indicates a mistake if he wrote \(x - 3 = 0\) instead of \(x + 3 = 0\).)
  • No, x minus 3 equals 0 in the second row should be x plus 3 equals 0. (This means the same as the previous response.)
  • No, he should have a third column where he should set x=0. (This is incorrect; it doesn't apply here as there should be no setting \(x=0\).)
  • Yes, the work is accurate. (This would mean no error, but if he wrote \(x - 3 = 0\) somewhere, this is not accurate.)
  • No, x+4=0 in the second row should be x−4=0. (This is incorrect; it should be \(x + 4 = 0\).)

Based on this analysis, the correct response to whether Oliver’s work is accurate is either of the first two responses, as he should have written \(x + 3 = 0\) and \(x + 4 = 0\).

The final conclusion is:

No, x+4=0 in the second row should be x−4=0 is incorrect as it should correctly be \( x + 4 = 0\), thus indicating Oliver’s work is not accurate if he stated it incorrectly in terms of signs.

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 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. 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. 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) or ​(x+2), must equal 0 for the entire function to equal 0.

Based on the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be equal to zero.

For the quadratic equation given:

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

According to the Zero Product Property, for this equation to hold true (i.e., for the entire expression to be equal to zero), at least one of the two factors must equal zero:

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

This leads to possible solutions for \(x\).

Now let's evaluate the provided responses:

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

    • This is incorrect. Only one factor needs to be zero for the product to be zero.

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

    • This is the same statement as the first and is incorrect for the same reason.

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

    • This is correct based on the Zero Product Property.

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

    • This is a repeat of the third option and is also correct.

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

    • This is incorrect; we know there are solutions.

  6. Neither of the factors, (2x−3) or (x+2), must equal 0 for the entire function to equal 0.

    • This is incorrect since at least one factor must be zero for the product to be zero.

Thus, the statements that must be true based on the Zero Product Property are:

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

This corresponds to options 3 and 4, both of which express the same valid condition.

pick one

Sure! The best choice is:

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

This statement correctly reflects the Zero Product Property.