SOLUTION: what natural can replace B to make perfect square trinomial 64X^2+BX+16

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Question 570907: what natural can replace B to make perfect square trinomial
64X^2+BX+16
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+f%28x%29+=+64x%5E2+%2B+b%2Ax+%2B+16+
The quadratic formula to find roots is:
x+=+%28+-b+%2B-+sqrt%28+b%5E2+-+4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+64+
+b+=+b+
+c+=+16+
To make +f%28x%29+ a perfect square, +b%5E2+-+4%2Aa%2Ac+
must be zero
+b%5E2+-+4%2Aa%2Ac++=+0+
+b%5E2+-+4%2A64%2A16+=+0+
+b%5E2+-+4096+=+0+
+b%5E2+=+4096+
+b+=+64+
+b+=+-64+
-----------
If +b+=+64+
+f%28x%29+=+64x%5E2+%2B+b%2Ax+%2B+16+
+f%28x%29+=+64x%5E2+%2B+64x+%2B+16+
+f%28x%29+=+4x%5E2+%2B+4x+%2B+1+
+f%28x%29+=+%28+2x+%2B+1+%29%5E2+
-----------
If +b+=+-64+
+f%28x%29+=+64x%5E2+-+64x+%2B+16+
+f%28x%29+=+4x%5E2+-+4x+%2B+1+
+f%28x%29+=+%28+2x+-+1+%29%5E2+
-----------
b can be replaced with either +64+ or +-64+
Here are plots of the 2 alternatives:
+graph%28+400%2C+400%2C+-4%2C+4%2C+-4%2C+8%2C+4x%5E2+%2B+4x+%2B+1%2C+4x%5E2+-+4x+%2B+1+%29+