document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #729198 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "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 $\"4x%5E2%2Bnx-7%29%5E2\"$ .

\n" ); document.write( "Each form gives an answer to the question.

\n" ); document.write( "(1) For the first form...

\n" ); document.write( " $\"%284x%5E2%2Bnx%2B7%29%5E2+=+16x%5E4%2B8nx%5E3%2B%28n%5E2%2B56%29x%5E2%2B14nx%2B49\"$

\n" ); document.write( "Then
\n" ); document.write( "8n = -24 --> n = -3
\n" ); document.write( "n^2+56 = 65 = a-1 --> a = 66
\n" ); document.write( "14n = -42 = b+1 --> b = -43

\n" ); document.write( "(2) For the second form...

\n" ); document.write( " $\"%284x%5E2%2Bnx-7%29%5E2+=+16x%5E4%2B8nx%5E3%2B%28n%5E2-56%29x%5E2-14nx%2B49\"$

\n" ); document.write( "Then
\n" ); document.write( "8n = -24 --> n = -3
\n" ); document.write( "n^2-56 = -47 = a-1 --> a = -46
\n" ); document.write( "-14n = 42 = b+1 --> b = 41

\n" ); document.write( "Answer: Two solutions
\n" ); document.write( "(1) a=66, b=-43
\n" ); document.write( "(2) a=-46, b=41 \n" ); document.write( "

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