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

Answer 1

To use the Zero Product Property to find the horizontal distance \( x \) when the height \( h(x) \) of the soccer ball is equal to 0, Leon has correctly set up the equation:

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

According to the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be zero. In this case, we have two factors:

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

Now, we can set each factor equal to zero:

1. From \(-\frac{1}{2} x = 0\): \[ x = 0 \quad \text{(the soccer ball is at ground level at the starting point)} \]

2. From \(x - 20 = 0\): \[ x = 20 \quad \text{(the next point at which the soccer ball will hit the ground again)} \]

Thus, the solutions to the equation \(0 = -\frac{1}{2} x (x - 20)\) are \(x = 0\) and \(x = 20\).

In conclusion, the soccer ball hits the ground at two points: when it has traveled 0 units (the point of kick) and after traveling 20 units horizontally. Therefore, the horizontal distance the ball has traveled when it hits the ground is 20 units.