Multiple choice

Solve the equation $x^2 - (\sqrt 3 + 1) x + \sqrt 3 = 0$ by the method of completing the square.

  1. $x = \sqrt 1, 3$
  2. $x = \sqrt 5, 1$
  3. $x = \sqrt 10, 2$
  4. $x = \sqrt 3, 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Rearranging and completing the square gives (x - (sqrt(3) + 1)/2)^2 = ((sqrt(3) - 1)/2)^2. Hence x = ((sqrt(3) + 1) +/- (sqrt(3) - 1))/2, giving x = 1 and x = sqrt(3).

AI explanation

Take half of the coefficient of x, which is -(sqrt(3) + 1)/2, and add and subtract its square to the equation. This forms the perfect square (x - (sqrt(3) + 1)/2)^2 = ((sqrt(3) + 1)/2)^2 - sqrt(3). Simplifying the right side gives (3 + 2sqrt(3) + 1)/4 - sqrt(3) = 1 + sqrt(3)/2 - sqrt(3) = 1 - sqrt(3)/2 = (2 - sqrt(3))^2 / 4. Taking the square root of both sides gives x - (sqrt(3) + 1)/2 = plus or minus (sqrt(3) - 1)/2, resulting in x = sqrt(3) and x = 1.

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