Multiple choice

Consider the equation (1 - x)4 + (5 - x)4 = 82. What is the number of real roots of the equation?

  1. 0

  2. 2

  3. 4

  4. 8
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let f(x) = (1-x)^4 + (5-x)^4. This is a convex function. The minimum occurs at the midpoint of 1 and 5, which is x = 3. f(3) = (-2)^4 + (2)^4 = 16 + 16 = 32. Since 32 < 82, the function crosses 82 at two points (one less than 3, one greater than 3).

AI explanation

Use the symmetry method by letting the average of the bases be y, so 1 - x = 3 - y and 5 - x = 3 + y. Substituting these into the equation yields (3 - y)^4 + (3 + y)^4 = 82, which expands to 2(y^4 + 24y^2 + 81) = 82. Simplifying gives y^4 + 24y^2 + 40 = 0, and treating this as a quadratic in y^2 yields two positive solutions for y^2. Each positive solution produces two real values for y, resulting in 2 real roots for x.

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