document.write( "Question 1077332: For what real values of c is 9x^2 + 16x + c the square of a binomial? You many find more than one. \n" ); document.write( "

Algebra.Com's Answer #691851 by MathLover1(20849) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " The exact answer can be had by letting $\"ax%2Bb\"$ be that $\"binomial\"$ , so that:\r
\n" ); document.write( "\n" ); document.write( " $\"9x%5E2+%2B+16x+%2B+c+=+%28ax+%2B+b%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"9x%5E2+%2B+16x+%2B+c+=+a%5E2x%5E2+%2B+2abx+%2B+b%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "For those to be $\"identical\"$ polynomials, all $\"corresponding\"$ $\"coefficients\"$ $\"must\"$ be $\"equal\"$ , so:\r
\n" ); document.write( "\n" ); document.write( " $\"9+=+a%5E2\"$ .... leads to $\"a=3\"$ or $\"a=-3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"16+=+2ab\"$ .... leads to $\"b+=+16%2F2a\"$ , $\"b+=+8%2Fa\"$ ; so, $\"b+=+8%2F3\"$ or $\"b=-8%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"c+=+b%5E2\"$ .... whether $\"c\"$ is + or - $\"8%2F3\"$ , the square is $\"64%2F9\"$ \r
\n" ); document.write( "\n" ); document.write( "So, $\"highlight%28c+=+64%2F9%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Another approach is to use the $\"discriminant\"$ . A quadratic is a perfect square if and only if its discriminant is zero. So:\r
\n" ); document.write( "\n" ); document.write( " $\"0+=+b%5E2+-+4ac+=+16%5E2-+4%289%29%28c%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"0+=+256+-+36c\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"c+=+256%2F36\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"highlight%28c+=+64%2F9%29\"$ \r
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