Multiple choice

Find the roots of each of the following quadratic equations by the method of completing the sqaures: $x^2 - 6x + 4 = 0$

  1. $ 9\pm \sqrt 8$
  2. $6 \pm \sqrt 7$
  3. $3 \pm \sqrt 5$
  4. $ 1\pm \sqrt 11$
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C Correct answer
Explanation

For x^2 - 6x + 4 = 0, complete the square: (x^2 - 6x + 9) - 9 + 4 = 0, which is (x-3)^2 - 5 = 0. Thus, (x-3)^2 = 5, so x-3 = +/- sqrt(5), giving x = 3 +/- sqrt(5).

AI explanation

Using the method of completing the squares, we move the constant in x squared minus 6x plus 4 equals 0 to get x squared minus 6x equals negative 4. We add the square of half the coefficient of x, which is (6 divided by 2) squared or 9, to both sides to form x squared minus 6x plus 9 equals 5. This allows us to factor the left side into a perfect square, giving (x minus 3) squared equals 5. Taking the square root of both sides and solving for x yields the roots x equals 3 plus the square root of 5 and x equals 3 minus the square root of 5.

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