If x, y, z satisfy:
x + y + z = 12,
1/x + 1/y + 1/z = 2, and
x3 + y3 + z3 = -480,
find x2y + xy2 + x2z + xz2 + y2z + yz2.
Wolfram Alpha solves the equation as the following values for x, y and z, and they have been placed into Matlab, as syms (symbolic values):
x = sym(2 *(2 + (2 *(sqrt(777) - 27))^(1/3)/3^(2/3) - (2* 2^(2/3))/(3* (sqrt(777) - 27))^(1/3)) )
y = sym(1/2 *(12 - 2 *(2 + (2 *(sqrt(777) - 27))^(1/3)/3^(2/3) - (2 *2^(2/3))/(3 *(sqrt(777) - 27))^(1/3)) - sqrt((2 *(2 + (2 *(sqrt(777) - 27))^(1/3)/3^(2/3) - (2* 2^(2/3))/(3* (sqrt(777) - 27))^(1/3)) - 12)^2 - 4 *(64 - 24* (2 + (2 *(sqrt(777) - 27))^(1/3)/3^(2/3) - (2* 2^(2/3))/(3* (sqrt(777) - 27))^(1/3)) + 4 *(2 + (2* (sqrt(777) - 27))^(1/3)/3^(2/3) - (2* 2^(2/3))/(3 *(sqrt(777) - 27))^(1/3))^2))) )
z = sym(4 - (2* (sqrt(777) - 27))^(1/3)/3^(2/3) + (2* 2^(2/3))/(3* (sqrt(777) - 27))^(1/3) + 1/2 *sqrt((2 *(2 + (2* (sqrt(777) - 27))^(1/3)/3^(2/3) - (2 *2^(2/3))/(3* (sqrt(777) - 27))^(1/3)) - 12)^2 - 4 *(64 - 24 *(2 + (2* (sqrt(777) - 27))^(1/3)/3^(2/3) - (2* 2^(2/3))/(3 *(sqrt(777) - 27))^(1/3)) + 4* (2 + (2 *(sqrt(777) - 27))^(1/3)/3^(2/3) - (2 *2^(2/3))/(3 *(sqrt(777) - 27))^(1/3))^2)))
Then Matlab is asked to show
(x^2)*y + x*(y^2) + (x^2)*z + x*(z^2) + (y^2)*z + y*(z^2)
which it does as
ans =
29972201546825169053303689227439180062402902955/44601490397061246283071436545296723011960832
which it evaluates as
>> eval(ans)
ans =
672
>
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Posted by Charlie
on 2025-03-29 15:28:35 |
a try
