Multiple choice

The number of roots of the equation $|x-x^{2}-2|=|2x-3-x^{2}|$ is

  1. $0$
  2. $1$
  3. $2$

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

The equation is |x - x^2 - 2| = |2x - 3 - x^2|. Since both sides are absolute values, we solve x - x^2 - 2 = 2x - 3 - x^2 or x - x^2 - 2 = -(2x - 3 - x^2). The first case gives -x - 2 = -3, so x = 1. The second case gives x - x^2 - 2 = -2x + 3 + x^2, leading to 2x^2 - 3x + 5 = 0, which has no real roots.

AI explanation

Let f(x) equal x squared minus x plus 2 and g(x) equal x squared minus 2x plus 3. The equation becomes the absolute value of negative f(x) equals the absolute value of negative g(x), so f(x) equals plus or minus g(x). Equating f(x) to g(x) yields x equals 1, and substituting 1 back shows f(1) equals 2 and g(1) equals 2, making it a valid root. Equating f(x) to negative g(x) yields a negative constant equaling zero, which has no solutions. The number of roots is 1.

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