f(f(x)) = x

Algebra Level 3

Find the sum of all roots of the following equation.

( x 2 − 3 x + 3 ) 2 − 3 ( x 2 − 3 x + 3 ) = ( x − 3 ) (x^2 -3x +3)^2 - 3(x^2 -3x +3) = (x-3)


The answer is 6.

Ossama Ismail by Ossama Ismail , 63, Egypt

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2 solutions

The given equation simplifies to

( ( x 2 − 3 x + 3 ) − 3 ) ( x 2 − 3 x + 3 ) = x − 3 ⟹ x ( x − 3 ) ( x 2 − 3 x + 3 ) = x − 3 ⟹ ((x^{2} - 3x + 3) - 3)(x^{2} - 3x + 3) = x - 3 \Longrightarrow x(x - 3)(x^{2} - 3x + 3) = x - 3 \Longrightarrow

( x − 3 ) ( x × ( x 2 − 3 x + 3 ) − 1 ) = 0 ⟹ ( x − 3 ) ( x 3 − 3 x 2 + 3 x − 1 ) = 0 ⟹ ( x − 3 ) ( x − 1 ) 3 = 0 (x - 3)(x \times (x^{2} - 3x + 3) - 1) = 0 \Longrightarrow (x - 3)(x^{3} - 3x^{2} + 3x - 1) = 0 \Longrightarrow (x - 3)(x - 1)^{3} = 0 .

So x = 3 x = 3 is a single root and x = 1 x = 1 is a triple root, making the sum of all roots 3 + 1 + 1 + 1 = 6 3 + 1 + 1 + 1 = \boxed{6} .

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There is no need to solve the equation to find the answer. The equation can be rewritten as x 4 − 6 x 3 + 12 x 2 − 10 x + 3 = 0 x^4-6x^3+12x^2-10x+3=0 ; the sum of its roots is equal to minus the coefficient of the second term.

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