The path of a soccer ball can be modeled by the equation h(x)=−1/2x(x−20)

Question

, 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)
Responses

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.

Solve the two equations: −1/2x=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 .

Cancel out the x and then multiply −1/2
and −20
.

Answer 1

To find the horizontal distance when the soccer ball hits the ground (when the height \( h(x) \) equals 0), Leon can use the Zero Product Property effectively. The equation he has 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 equal zero. In this case, we have two factors: \(-\frac{1}{2}x\) and \(x - 20\).

To apply the Zero Product Property, Leon needs to set each factor equal to zero:

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

Now, let's solve each equation:

1. From \(-\frac{1}{2}x = 0\), multiplying both sides by \(-2\) gives \(x = 0\).

2. From \(x - 20 = 0\), adding 20 to both sides gives \(x = 20\).

Thus, the values for \(x\) that satisfy the equation are \(x = 0\) and \(x = 20\). This means the soccer ball has traveled 0 units when it is kicked (at the start) and 20 units when it hits the ground.

So, the correct response is:

Solve the two equations: −1/2x=0 and x−20=0.