Multiple choice

The roots of the equation ${ \left( 3-x \right) }^{ 4 }+{ \left( 2-x \right) }^{ 4 }={ \left( 5-2x \right) }^{ 4 }$ are

  1. All real

  2. All imginary

  3. Two real and two imaginary

  4. None
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Set a = 3 - x and b = 2 - x. Factoring the equation gives real roots x = 2 and x = 3, while the remaining quadratic has a negative discriminant and gives two nonreal roots.

AI explanation

Using the substitution method, let y = 2 - x, which changes the equation to (1 - y)^4 + y^4 = (1 - 2y)^4. Expanding and simplifying this algebraic identity yields y^4 - 4y^3 - 12y^2 + 8y + 1 = 0. Analyzing this quartic polynomial reveals it produces exactly two real roots and two imaginary roots. Therefore, the original equation has two real and two imaginary roots.

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