SOLUTION: Find the values of a and b if 16x^4-24x^3+(a-1)x^2+(b+1)x+49 is a perfect square.

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Question 1114140: Find the values of a and b if 16x^4-24x^3+(a-1)x^2+(b+1)x+49 is a perfect square.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Since the leading term is 16x^4 and the constant term is 49, the perfect square must be of the form %284x%5E2%2Bnx%2B7%29%5E2 or 4x%5E2%2Bnx-7%29%5E2.

Each form gives an answer to the question.

(1) For the first form...

%284x%5E2%2Bnx%2B7%29%5E2+=+16x%5E4%2B8nx%5E3%2B%28n%5E2%2B56%29x%5E2%2B14nx%2B49

Then
8n = -24 --> n = -3
n^2+56 = 65 = a-1 --> a = 66
14n = -42 = b+1 --> b = -43

(2) For the second form...

%284x%5E2%2Bnx-7%29%5E2+=+16x%5E4%2B8nx%5E3%2B%28n%5E2-56%29x%5E2-14nx%2B49

Then
8n = -24 --> n = -3
n^2-56 = -47 = a-1 --> a = -46
-14n = 42 = b+1 --> b = 41

Answer: Two solutions
(1) a=66, b=-43
(2) a=-46, b=41