Multiple choice

The x-coordinates of the vertices of a square are the roots of the $x^{2} - 3|x| + 2 = 0$ and the y-coordinates of the vertices are the roots of the equation $y^{2} - 3y=0$ the possible vertices of the square is/ are

  1. $(1, 0), (2, 3), (2, 0), (1, 3)$
  2. $(-1, 3), (-2, 3), (-2, 0), (-1, 0)$
  3. $(2, 0), (1, 3), (-1, 0), (2, 0)$
  4. $(-2, 0), (-2, 3), (1, 0), (1, 3)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The roots of x^2 - 3|x| + 2 = 0 are x = 1, 2, -1, -2. The roots of y^2 - 3y = 0 are y = 0, 3. The vertices are formed by combinations of these coordinates. Option D correctly lists four points that form a square with side length 3.

AI explanation

Solving the equation for the x-coordinates, x squared minus 3 times the absolute value of x plus 2 equals 0 gives absolute value of x equals 1 or 2, meaning the x-coordinates are 1, -1, 2, and -2. Solving the equation for the y-coordinates, y squared minus 3 times y equals 0 gives y equals 0 or 3. A square requires all four vertices to share exactly two distinct x-coordinates and two distinct y-coordinates, and the side lengths must be equal. The set of vertices (-2, 0), (-2, 3), (1, 0), and (1, 3) uses only the x-coordinates -2 and 1, the y-coordinates 0 and 3, forming side lengths of 3 and 3, which perfectly constitutes a square. Therefore, the possible vertices are (-2, 0), (-2, 3), (1, 0), and (1, 3).

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