document.write( "Question 661685: the equation x^2-x+1=0 has zeros p and q, and equation 3x^2-2x+3= 0 has zeros r and s
\n" ); document.write( "without solving the equations compute (p-r)(q-r)(p-s)(q-s) \n" ); document.write( "

Algebra.Com's Answer #411777 by kevwill(135) $\"\"$ $\"About$

You can put this solution on YOUR website!
There might be a shorter way to do this, but here's how I solved it.
\n" ); document.write( "Given that $\"x%5E2+-+x+%2B+1+=+0\"$ has roots at x=p and x=q, we know that
\n" ); document.write( " $\"x%5E2+-+x+%2B+1+=+0\"$ can be factored into $\"%28x+-+p%29%28x+-+q%29+=+0\"$ , so equivalently
\n" ); document.write( " $\"x%5E2+-+x+%2B+1+=+%28x+-+p%29%28x+-+q%29+=+x%5E2+-%28p+%2B+q%29%2Ax+%2B+%28p+%2A+q%29+=+0\"$ .
\n" ); document.write( "This gives us $\"p+%2B+q+=+1\"$ and $\"p+%2A+q+=+1\"$
\n" ); document.write( "Similarly, $\"3%2Ax%5E2+-+2%2Ax+%2B+3+=+0\"$ can be factored into $\"3%28x+-+r%29%28x+-+s%29+=+0\"$ .
\n" ); document.write( "We can divide both sides of each by 3 to get
\n" ); document.write( "

.
\n" ); document.write( "This gives us $\"r+%2B+s+=+2%2F3\"$ and $\"r+%2A+s+=+1\"$
\n" ); document.write( "The next thing we need is:
\n" ); document.write( " $\"%28p+%2B+q%29%5E2+=+p%5E2+%2B+2pq+%2B+q2+=+%28p%5E2+%2B+q%5E2%29+%2B+2%2Ap%2Aq\"$
\n" ); document.write( " $\"%28p+%2B+q%29%5E2+-+2%2Ap%2Aq+=+p%5E2+%2B+q%5E2\"$
\n" ); document.write( "And we know from above that $\"p%2Bq+=+1\"$ and $\"p%2Aq+=+1\"$ so
\n" ); document.write( " $\"1%5E2+-+2%2A1+=+p%5E2+%2B+q%5E2\"$ or
\n" ); document.write( " $\"p%5E2+%2B+q%5E2+=+1+-+2+=+-1\"$
\n" ); document.write( "Similarly, we can get
\n" ); document.write( "

\n" ); document.write( "Finally, we need one last step:
\n" ); document.write( " $\"%28p+%2B+q%29%28r+%2B+s%29+=+pr+%2B+ps+%2B+qr+%2B+qs\"$
\n" ); document.write( "But since $\"p%2Bq+=+1\"$ and $\"r%2Bs+=+2%2F3\"$ we have
\n" ); document.write( " $\"pr+%2B+ps+%2B+qr+%2B+qs+=+1%2A%282%2F3%29+=+2%2F3\"$
\n" ); document.write( "Now we can multiply out (p-r)(q-r)(p-s)(q-s)
\n" ); document.write( "Let's regroup:
\n" ); document.write( "

\n" ); document.write( "

\n" ); document.write( "Substituting pq=1 and rs=1:
\n" ); document.write( "

\n" ); document.write( " $\"%28p-r%29%2A%28q-r%29%2A%28p-s%29%2A%28q-s%29+=+%282+-+ps+-+qr%29%282+-+qs+-+pr%29\"$
\n" ); document.write( "

\n" ); document.write( "Again, substituting pq=1 and rs=1:
\n" ); document.write( "

\n" ); document.write( "

\n" ); document.write( "From above, $\"pr+%2B+ps+%2B+qr+%2B+qs+=+2%2F3\"$ , $\"p%5E2+%2B+q%5E2+=+-1\"$ , and $\"r%5E2+%2B+s%5E2+=+-14%2F9\"$
\n" ); document.write( "

\n" ); document.write( " $\"%28p-r%29%2A%28q-r%29%2A%28p-s%29%2A%28q-s%29+=+4+-+4%2F3+-+1+-14%2F9\"$
\n" ); document.write( " $\"%28p-r%29%2A%28q-r%29%2A%28p-s%29%2A%28q-s%29+=+36%2F9+-+12%2F9+-+9%2F9+-14%2F9\"$
\n" ); document.write( " $\"%28p-r%29%2A%28q-r%29%2A%28p-s%29%2A%28q-s%29+=+%2836+-+12+-+9+-+14%29%2F9\"$
\n" ); document.write( " $\"%28p-r%29%2A%28q-r%29%2A%28p-s%29%2A%28q-s%29+=+1%2F9\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );