document.write( "Question 977117: prove that the equation mx(x^2+2x+3)=x^2-2x-3 has exactly one real root if m=1 and exactly 3 real roots if m=-2/3 \n" ); document.write( "

Algebra.Com's Answer #600460 by anand429(138) $\"\"$ $\"About$

You can put this solution on YOUR website!
Simplifying, our equation becomes,
\n" ); document.write( " $\"mx%5E3+%2B+%282m-1%29x%5E2+%2B+%283m%2B2%29x+%2B+3+=0\"$
\n" ); document.write( "For m =1,
\n" ); document.write( "we get,
\n" ); document.write( " $\"x%5E3+%2B+x%5E2+%2B+5x+%2B+3+=0\"$
\n" ); document.write( "Comparing with standard form,
\n" ); document.write( " $\"ax%5E3%2Bbx%5E2%2Bcx%2Bd+=+0\"$
\n" ); document.write( "Let $\"f+=+%28%283c%2Fa%29-%28b%5E2%2Fa%5E2%29%29%2F3\"$
\n" ); document.write( "=> $\"f=+%2815+-1%29%2F3\"$
\n" ); document.write( "=> $\"f+=+14%2F3\"$
\n" ); document.write( " Let $\"g=+%28%282b%5E3%2Fa%5E3%29-%289bc%2Fa%5E2%29+%2B+%2827d%2Fa%29%29%2F27\"$
\n" ); document.write( "=> $\"g=+%282-45%2B81%29%2F27\"$
\n" ); document.write( "=> $\"g=38%2F27\"$ \r
\n" ); document.write( "\n" ); document.write( "Now, let $\"h=+%28g%5E2%2F4%29+%2B+%28f%5E3%2F27%29\"$
\n" ); document.write( "=> $\"h=+%2838%2F27%29%5E2%2F4+%2B+%2814%2F3%29%5E3%2F27\"$
\n" ); document.write( "Clearly, h>0
\n" ); document.write( "hence, there is only one real root.\r
\n" ); document.write( "\n" ); document.write( "Similarly, put m = -2/3 and solve as above,
\n" ); document.write( "If h<0, it means all three roots are real. \n" ); document.write( "

\n" );