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 traveled when the height of the soccer ball equals 0?(1 point)

Question

Responses

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 .

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 .

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.

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

Answer 1

To find the horizontal distance the ball has traveled when the height of the soccer ball equals 0, Leon can use the Zero Product Property. The equation he set up is:

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

According to the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be zero. This leads to two equations:

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

Now, we can solve each equation:

1. From \( -\frac{1}{2} x = 0 \), we get \( x = 0 \).
2. From \( x - 20 = 0 \), we get \( x = 20 \).

Thus, the horizontal distances where the height is 0 are \( x = 0 \) and \( x = 20 \).

The appropriate response is:

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

(Note: There is a small misprint in the version provided; it should state \( -\frac{1}{2}x = 0 \) instead of \( -12x = 0 \).)