Question 1209301
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Answer: <font color=red>-23/9</font>
This is approximately equal to -2.55556 where the 5's go on forever but we have to round at some point.



Explanation


I'll use x in place of t.


x^2 -9x - 36 + 8x^2 + 55x + 41
= (x^2 + 8x^2) + (-9x + 55x) + ( -36 + 41)
= 9x^2 + 46x + 5


The vertex of y = ax^2+bx+c is located at (h,k) where h = -b/(2a)
y = 9x^2 + 46x + 5 has the coefficients a = 9, b = 46, c = 5


h = -b/(2a)
h = -46/(2*9)
h = -46/18
h = (-2*23)/(2*9)
h = <font color=red>-23/9</font>
This is the x coordinate of the vertex. 
This x value will make 9x^2 + 46x + 5 as small as possible.
Note that a = 9 is positive, so the parabola opens upward, which places the vertex at the lowest point.


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Another approach would be to use the quadratic formula to solve 9x^2 + 46x + 5 = 0 to get the roots x = -5 and x = -1/9
I'll leave the scratch work for the student to do.
Or you could factor 9x^2 + 46x + 5 into (x+5)(9x+1) to be able to see the roots easily. 


x = -5 and x = -1/9 are where the parabola crosses the x axis.
The midpoint of the x intercepts is the x coordinate of the vertex.
This is due to the parabola's mirror symmetry about the center line. 


p,q are the roots
h = x coordinate of the vertex (h,k)
h = (1/2)*(p+q)
h = (1/2)*( -5 + (-1/9) )
h = (1/2)*( -45/9 + (-1/9) )
h = (1/2)*(-46/9)
h = (-46)/(2*9)
h = (-2*23)/(2*9)
h = <font color=red>-23/9</font>
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