What is the number of real roots of the equation?
Consider the equation (1-x)^{4}+(5-x)^{4}=82.
- A. 0
- B. 2 ✓
- C. 4
- D. 8
Correct Answer: B. 2
Explanation
Substitute y = 3-x to make the equation symmetric. This gives (y-2)^4 + (y+2)^4 = 82. Expanding using the binomial theorem yields 2(y^4 + 24y^2 + 16) = 82, which simplifies to y^4 + 24y^2 - 25 = 0. Factoring gives (y^2 + 25)(y^2 - 1) = 0. For real roots, y^2 = 1 \implies y = \pm 1. Thus, x = 2 or x = 4. There are exactly 2 real roots.
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