Question

1
Which should be added to
make x4 + 64 a perfect square?​

Answer 1

The two most obvious answers are  −4

−x4 and  −4−4

. In the first case the expression becomes  22

22

, in the second case it becomes  (2)2

(x2)2

Of course you could add  −42−4x2

 to get  (2–4)2

(x2–4)2

 or you could add  42

4x2

 to get  (2+4)2

(x2+4)2

.A more creative solution would be to add  1168

116x8

 to get  (144+2)2

(14x4+2)2

But why not add  −(4+4)−(x4+4 )

After all,  0

0  is a perfect square for all  x

There are an infinite number of other solutions, so there is no “should be added”, there is only “could be added”