document.write( "Question 269918: Given that r and s are roots of the quadratic equation 3(x^2)+1=7x, find (r^3)s+r(s^3), without solving for the roots of the original equation. \n" ); document.write( "

Algebra.Com's Answer #197770 by drk(1908) $\"\"$ $\"About$

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step 1 - rewrite in descending order = 0 to get
\n" ); document.write( " $\"3x%5E2+-+7x+%2B+1+=+0\"$
\n" ); document.write( "step 2 - the sum of the roots = -b/a or
\n" ); document.write( "r + s = 7/3
\n" ); document.write( "step 3 - the product of the roots = c/a or
\n" ); document.write( "rs = 1/3.
\n" ); document.write( "step 4 - we want r^3s + s^3r. Factoring, we get
\n" ); document.write( "rs(s^2 + r^2)
\n" ); document.write( "step 5- we know rs = 1/3. squaring (r+s), we get
\n" ); document.write( "(r+s)^2 = (7/3)^2
\n" ); document.write( "r^2 + 2rs + s^2 = 49/9
\n" ); document.write( "step 6 - since rs = 1/3, 2rs = 2/3, we get
\n" ); document.write( "r^2 + 2/3 + s^2 = 49/9
\n" ); document.write( "step 7 - subtract 2/3 to get
\n" ); document.write( "r^2 + s^2 = 49/9 - 2/3 = 43/9
\n" ); document.write( "step 8 - from step 5, we get
\n" ); document.write( "(1/3)(43/9) = 43/ 27. \n" ); document.write( "

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