document.write( "Question 1041380: The quadratic equation $\"2x%5E2+%2B+4x+%2B+p=0\"$ has roots $\"alpha\"$ and $\"beta\"$ , and the equation $\"x%5E2%2B4x%2B6+=+0\"$ has roots $\"k%2Falpha\"$ and $\"k%2Fbeta\"$ . Find the values of k and p. \n" ); document.write( "

Algebra.Com's Answer #656404 by robertb(5830) $\"\"$ $\"About$

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$\"2x%5E2+%2B+4x+%2B+p=0\"$ <==> $\"x%5E2+%2B+2x+%2B+p%2F2=0\"$ \r
\n" ); document.write( "\n" ); document.write( "==> $\"-2+=+alpha+%2B+beta\"$ and $\"alpha%2Abeta+=+p%2F2\"$ .\r
\n" ); document.write( "\n" ); document.write( "Now, $\"x%5E2%2B4x%2B6+=+0\"$ ==> $\"-4+=+k%2Falpha+%2B+k%2Fbeta\"$ \r
\n" ); document.write( "\n" ); document.write( "<==> $\"-4+=+%28k%28alpha%2Bbeta%29%29%2F%28alpha%2Abeta%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "==> $\"-4+=+%28k%2A-2%29%2F%28p%2F2%29\"$ ==> k = p, after clearing fractions and simplifying.\r
\n" ); document.write( "\n" ); document.write( "Also, from the given, $\"6++=%28k%2Falpha%29%2A%28k%2Fbeta%29+=+k%5E2%2F%28alpha%2Abeta%29\"$ \r
\n" ); document.write( "\n" ); document.write( "==> $\"6%2Aalpha%2Abeta+=+k%5E2\"$ ==> $\"6%2A%28p%2F2%29+=+k%5E2%7D%0D%0A%0D%0A==%3E+%7B%7B%7B3p+=+k%5E2\"$ ===> $\"3p+=+p%5E2\"$ ===> p = 0,3.\r
\n" ); document.write( "\n" ); document.write( "p = 0 is not acceptable because it will imply a root of 0 in $\"2x%5E2+%2B+4x+%2B+p=0\"$ and an undefined root in $\"x%5E2%2B4x%2B6+=+0\"$ .\r
\n" ); document.write( "\n" ); document.write( "Therefore, $\"highlight%28k+=+p+=+3%29\"$ .\r
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