The path of a soccer ball can be modeled by the equation h (x) = -1/2x (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 = -1/2x (x - 20). How can he use the Zero Product Property to find the horizontal distance the ball has the ball has traveled when the height of the soccer ball equals 0?

Question

Cancel out the x and then multiply -1/2 and -20.
Solve the two equations: -1/2x = 0 and x - 20 = 0.
He cannot use the Zero Product Property because the height will never be 0.
Plug in 0 for all the x terms in the function.

Answer 1

To solve the equation \( 0 = -\frac{1}{2}x(x - 20) \) using the Zero Product Property, we start by understanding that this property states that if a product of multiple factors equals zero, then at least one of the factors must be zero.

In this case, we have two factors in the equation:

1. \(-\frac{1}{2}x\)
2. \(x - 20\)

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

1. \(-\frac{1}{2}x = 0\)
2. \(x - 20 = 0\)

Now, we solve each equation separately.

For the first equation:

\[ -\frac{1}{2}x = 0 \]

To solve for \(x\), we can multiply both sides by \(-2\):

\[ x = 0 \]

For the second equation:

\[ x - 20 = 0 \]

Adding \(20\) to both sides gives:

\[ x = 20 \]

Now we have two solutions: \(x = 0\) and \(x = 20\).

The \(x = 0\) solution corresponds to the point when the ball is kicked (starting point), and \(x = 20\) corresponds to the point when the ball hits the ground.

Thus, the ball travels a horizontal distance of 20 units before it hits the ground.

In summary, the correct approach is to set the factors to zero and solve, leading us to the horizontal distances the ball has traveled when its height equals 0. The proper choice would be to "solve the two equations: \(-\frac{1}{2}x = 0\) and \(x - 20 = 0\)."