Korean Mathematics Competition [2000]
Algebra
Level
3
Find the sum of all real numbers satisfying the equation above.
The answer is 0.
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1 solution
Relevant wiki: Real Numbers
2 x + 3 x − 4 x + 6 x − 9 x = 1
Setting 2 x = a and 3 x = b , The Equation becomes 1 + a 2 + b 2 − a − b − a b = 0 Multiplying both sides of the last equation by 2 and completing the squares gives: ( 1 − a ) 2 + ( a − b ) 2 + ( b − 1 ) 2 = 0 Therefore 1 = 2 x = 3 x , so x = 0 is the only solution