Multiple choice

The sum of all roots of the roots of the equations $\displaystyle |x-1|^{2}-5|x-1|+6=0$ is

  1. 5

  2. 4

  3. 2

  4. 6
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B Correct answer
Explanation

Let y = |x-1|. The equation is y^2 - 5y + 6 = 0, so (y-2)(y-3) = 0. Thus |x-1| = 2 or |x-1| = 3. This gives x-1 = 2, -2, 3, -3, so x = 3, -1, 4, -2. The sum is 3 - 1 + 4 - 2 = 4.

AI explanation

Let y equal the absolute value of the quantity x minus 1, which changes the given equation into y^2 - 5y + 6 = 0. Factoring this quadratic gives (y - 2)(y - 3) = 0, so y equals 2 or 3. Substituting the absolute value expression back gives two equations: the absolute value of x minus 1 equals 2, and the absolute value of x minus 1 equals 3. Solving these yields x = 3, x = -1, x = 4, and x = -2, and adding all four roots gives a sum of 4.

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