Question 957751: The height of a baseball (h) in meters (t) seconds after it has been tossed out a window is given by the equation h= -5t^2 + 20t + 15. A boy shoots at the baseball with a trajectory of h= 3t +3. Will the paintball hit the baseball? If so at what height and time?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! yes it will.
it will intersect with the ball at (x,y) = (4,15).
you can easily see that from the graph of both equations shown below:
this can be solved algebraically as follows:
i used x instead of t and i used y instead of h.
this was mostly for graphing purposes.
the equations are the same otherwise.
the path of the ball is y = -5x^2 + 20x + 15.
the path of the projectile that the boy shoots is y = 3x + 3.
the projectile will hit the ball when these equations intersect.
you need to solve these two equations simultaneously to find the intersection point.
the two equations are:
y = 3x + 3 and y = -5x^2 + 20x + 15.
since y = 3x + 3 in the first equation, you can replace y with 3x + 3 in the second equation to get:
3x + 3 = -5x^2 + 20x + 15.
subtract 3x and subtract 3 from both sides of the equation and you will get 0 = -5x^2 + 20x + 15 - 3x - 3.
combine like terms to get 0 = -5x^2 + 17x + 12 which can also be shown as -5x^2 + 17x + 12 = 0
your equation to be factored is -5x^2 + 17x + 12 = 0.
in order to factor this equation using the box method or the split the middle term method, it is necessary to factor out the greatest common factor and to make the x^2 term positive.
i'll be using the splitting the middle term method.
multiply both sides of this equation by -1 to get 5x^2 - 17x - 12 = 0.
there is no greatest common factor so the equation you will be factoring is 5x^2 - 17x - 12 = 0
multiply the coefficient of the x^2 term by the constant term to get 60.
the sign is not important for this part of the process.
get all possible factors of 60 to get 1*60, 2*30, 3*20, 4*15, 5*12, 6*10.
since your constant term is negative and your x term is negative, you are looking for factors that will add up to a difference of 17.
20*3 will do it because 20 - 3 = 17 and 3 - 20 = -17
since your middle term is -17, then 3 - 20 are the factors you want.
you now split the middle term into two pieces.
-17x becomes 3x - 20x.
your equation to be factored becomes 5x^2 + 3x - 20x - 12 = 0
group the x^2 term and the first x term together and group the second x term and the constant term together to get:
(5x^2 + 3x) - (20x + 12) = 0
not that -20x - 12 becomes - (20x + 12) when you group the terms together.
factor both of these grouped terms to get:
x * (5x + 3) - 4 * (5x + 3) = 0
when you factor the grouped terms, you are looking for a common factor between them. in this case, the common term is 5x + 3.
factor out the common term of (5x + 3) to get:
x * (5x + 3) - 4 * (5x + 3) = 0 becomes:
(x - 4) * (5x + 3) = 0
those are your factors.
set each of these factors to 0 and you will get:
x = 4 or x = -3/5.
x = -3/5 is invalid because x can't be negative.
your solution will be at x = 4.
when x = 4, the equation of y = 5x^2 - 17x - 12 is equal to 0.
go back to your original two equations before you did anything with them.
they are:
y = 3x + 3 and y = -5x^2 + 20x + 15.
replace x with 4 in each equation.
y = 3x + 3 becomes y = 15.
y = -5x^2 + 20x + 15 becomes y = -80 + 80 + 15 which becomes y = 15.
the two equations will intersect when x = 4 at the point (4,15).
if you could not find the roots using the factoring method, then you would have been able to find them using the quadratic formula.
unless the factoring method give you a clear picture of what the factors should be, it is always recommended to go to the quadratic formula because that formula will always find you the roots, whether they are real or not.
the quadratic formula is 
the graph of the equation y = 5x^2 - 17x - 12 = 0 and the graph of the equation y = -5x^2 + 17x + 12 = 0 are shown below.
this is just to show you that the solution would have been the same regardless if the x^2 term was positive or negative.
the equations are flip flops of each other, but the roots are the same.
you can factor without making the x^2 term positive, but not by using the box method or the splitting the middle term method. those method require factoring out the greatest common factor and making the x^2 term positive.
here's the graph:
a good tutorial on factoring quadratic equations can be found here:
http://www.purplemath.com/modules/factquad.htm
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