Multiple choice

If the sum of the roots of the equation x2 - k2 x + 30kx - 161x - 64 = 0 is zero, then what is the difference of the roots?

  1. 15

  2. 16

  3. 17

  4. 18
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B Correct answer
Explanation

For the quadratic x^2 - (k^2 - 30k + 161)x - 64 = 0, the sum of roots is zero, meaning the coefficient of x must be zero. Thus, k^2 - 30k + 161 = 0. The roots of this quadratic are (x - a)(x + a) = 0, so the roots are a and -a. The product of roots is -a^2 = -64, so a = 8. The difference of roots is a - (-a) = 2a = 16.

AI explanation

First, group the x terms in the equation to get x2 + (30k - k2 - 161)x - 64 = 0. The sum of the roots is zero, so the coefficient of x must be zero, giving the equation 30k - k2 - 161 = 0. Multiplying by -1 and rearranging gives k2 - 30k + 161 = 0, which factors as (k - 7)(k - 23) = 0, yielding k = 7 or k = 23. Using k = 7, the original equation becomes x2 + (210 - 49 - 161)x - 64 = 0, which simplifies to x2 - 64 = 0. The roots are 8 and -8, making their difference 8 - (-8) = 16. (Using k = 23 yields the same absolute difference.)

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