SOLUTION: If t is a real number, what is the maximum possible value of the expression -t^2 + 8t -4 - 5t^2 + 14t + 55?

Judging by other problems I think you've posted, I think
the way you want it done is by completing the square:





Factor -6 out of the first two terms





Complete the square inside the parentheses:

1. Multiply the coefficient of t by 1/2
    
2. Square what you got:
    
3. Add and subtract that inside the parentheses
 


Factor the first three terms in the parentheses:



Distribute the -6, leaving the square of a binomial intact:



We want to make this as large as possible, so we make the
only non-positive term as small as possible.  Since it's a 
squared term the greatest (smallest amount of subtraction) 
it can be is 0, so the expression will be largest when 
t = 11/6.  Substituting:









   <---answer

Edwin

I still had not corrected every step in my solution above, but now it
is completely correct.

Edwin