Question

During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall of the opponent. When fired from the slingshot, the balloon will follow the path h (x) = -x^2 + 12x, where h (x) is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0 = -x^2 + 12x. How far will the balloon travel before it reaches the ground?

Answer 1

To determine how far the water balloon will travel before it reaches the ground, we need to find the values of \( x \) that satisfy the equation:

\[ 0 = -x^2 + 12x \]

Rearranging this equation gives:

\[ -x^2 + 12x = 0 \]

Factoring out \( x \) from the equation, we get:

\[ x(-x + 12) = 0 \]

This gives us two factors, leading to two possible solutions:

1. \( x = 0 \)
2. \( -x + 12 = 0 \)

Solving the second equation:

\[ -x + 12 = 0 \implies x = 12 \]

Thus, the solutions to the equation are \( x = 0 \) and \( x = 12 \). The \( x = 0 \) solution corresponds to the moment the balloon is fired, while \( x = 12 \) represents the distance at which the balloon reaches the ground.

Therefore, the balloon will travel 12 feet before it reaches the ground.