Question 1172154
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Consider the perfect square trinomial formula
(a+b)^2 = a^2 + 2ab + b^2


The expression 64x^2+56x+k has three terms, as does the right hand side of the equation above. 


Equating the two expressions shows that
64x^2 = a^2
56x = 2ab
k = b^2


To find k, we'll need to figure out b. 
To figure out b, we need to find 'a' first


If 64x^2 = a^2, then
a^2 = 64x^2
a^2 = (8x)^2
a = 8x
where the last step has us apply the square root to both sides. 
We could end up with a = -8x, but this value will ultimately lead to the same value of k. So we'll stick to a = 8x to make things simple.


Use that value of 'a', and the second equation we formed, to get
56x = 2ab
56x = 2(8x)b
56x = 16xb
56 = 16b
16b = 56
b = 56/16
b = (8*7)/(8*2)
b = 7/2
Note: if you went with a = -8x, then b = -7/2. Otherwise, b is positive.


Now we can compute k
k = b^2
k = (7/2)^2
k = (7^2)/(2^2)
k = 49/4


This means 64x^2+56x+k updates to {{{64x^2+56x+49/4}}} which factors to {{{(8x+7/2)^2}}}. 
Use the perfect square trinomial formula template, given at the very top of the solution, to help factor.
Because we can rewrite {{{64x^2+56x+49/4}}} into the form (expression)^2, this proves {{{64x^2+56x+49/4}}} is a perfect square trinomial. 


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Answer: 49/4
In decimal form, this is exactly 49/4 = 12.25
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