Multiple choice

Consider all the real roots of the equation x4 - 10x2 + 9 = 0. What is the sum of the absolute values of the roots?

  1. 4

  2. 6

  3. 8

  4. 10

  5. NA
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C Correct answer
Explanation

x^4 - 10x^2 + 9 = 0. Let y = x^2, so y^2 - 10y + 9 = 0. (y - 9)(y - 1) = 0. So x^2 = 9 or x^2 = 1. The roots are 3, -3, 1, -1. The absolute values are 3, 3, 1, 1. Sum = 3 + 3 + 1 + 1 = 8.

AI explanation

Treating x4 - 10x2 + 9 = 0 as a quadratic in x2, let y = x2, so y2 - 10y + 9 = 0. Factoring this gives (y - 1)(y - 9) = 0, meaning the values for y are 1 and 9, which correspond to x2 = 1 and x2 = 9. The real roots for x are therefore 1, -1, 3, and -3. The sum of their absolute values is |1| + |-1| + |3| + |-3|, which equals 8.

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