During a water balloon fight louisa used 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 solve the quadratic equation given by:

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

First, we can rearrange this into a standard form:

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

Factoring out an \(x\):

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

This gives us two solutions:

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

Solving for \(x\) in the second equation:

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

Thus, the two values of \(x\) where the height \(h(x)\) is zero are \(x = 0\) (the point of launch) and \(x = 12\) (the point where the balloon hits the ground).

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