Question 473676
The trajectory of the ball is modeled on the equation y = {{{-x^2/2}}} + 3x + 8
this means y = the height of the ball at distance x from the house
:
When the ball strikes the ground, h = 0, find x, therefore
{{{-x^2/2}}} + 3x + 8 = 0
we can make this easy to factor by multiplying equation by -2, results
x^2 - 6x - 16 = 0
Factors easily to
(x-8)(x+2) = 0
the positive solution is all we want here
x = 8 units is the distance from the house
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A graph of the original equation will illustrate this:
{{{ graph( 300, 200, -4, 12, -4, 16, (-x^2/2)+3x+8) }}} 
note that 
when x=0 the ball is at height of 8 which is where it is thrown from the house
when y=0 the ball strikes the ground 8 units from the house