Difference between revisions of "2002 Indonesia MO Problems/Problem 3"
Rockmanex3 (talk | contribs) (Solution to Problem 3) |
Rockmanex3 (talk | contribs) m (→Solution) |
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Factor the polynomial to get | Factor the polynomial to get | ||
<cmath>(a-2)^3 = 0</cmath> | <cmath>(a-2)^3 = 0</cmath> | ||
− | Since <math>a = 2</math> is a triple root to the polynomial, the only solution to the system of equations is <math>\boxed{(2,2,2)}</math>. | + | Since <math>a = 2</math> is a triple root to the polynomial, the only solution to the system of equations is <math>\boxed{(2,2,2)}</math>, and plugging the values back in satisfies the system. |
==See Also== | ==See Also== |
Revision as of 19:54, 14 July 2018
Problem
Find all real solutions from the following system of equations:
Solution
Square the first equation to get
Subtract the second equation from the result to get
Multiply the second equation by the first equation to get
Subtract the third equation and factor to get
Cube the first equation to get
If
,
, and
, the solution triplet is the roots of the polynomial
Factor the polynomial to get
Since
is a triple root to the polynomial, the only solution to the system of equations is
, and plugging the values back in satisfies the system.