SOLUTION: 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.

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.
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
step 1 - rewrite in descending order = 0 to get

step 2 - the sum of the roots = -b/a or
r + s = 7/3
step 3 - the product of the roots = c/a or
rs = 1/3.
step 4 - we want r^3s + s^3r. Factoring, we get
rs(s^2 + r^2)
step 5- we know rs = 1/3. squaring (r+s), we get
(r+s)^2 = (7/3)^2
r^2 + 2rs + s^2 = 49/9
step 6 - since rs = 1/3, 2rs = 2/3, we get
r^2 + 2/3 + s^2 = 49/9
step 7 - subtract 2/3 to get
r^2 + s^2 = 49/9 - 2/3 = 43/9
step 8 - from step 5, we get
(1/3)(43/9) = 43/ 27.
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