Question

A student solved the equation x^2 - 6x + 6 = 0 by completing the square.
Move numbers to the lines to complete the student's calculation.
x^2 - 6x + 6 = 0
(x^2 - 6x+___)+6=0
(x - 3)^2 =
x-3= + _____
x=___+and x=___+___

Answer 1

To solve the equation \( x^2 - 6x + 6 = 0 \) by completing the square, let's fill in the lines step by step.

1. Start with the original equation:
\[
x^2 - 6x + 6 = 0
\]

2. Complete the square. To do this, we need to find the value that completes the square. This value is \(\left(\frac{-6}{2}\right)^2 = 9\).

Now, we can rewrite the left side of the equation:
\[
(x^2 - 6x + 9) + 6 - 9 = 0
\]
or equivalently,
\[
(x^2 - 6x + 9) + (-3) = 0
\]
which simplifies to:
\[
(x - 3)^2 - 3 = 0
\]

This shows that we can fill in these blanks:
\[
(x^2 - 6x + 9) + 6 - 9 = 0
\]

Thus,
\[
(x^2 - 6x + \underline{9}) + \underline{-3} = 0
\]

3. Now we can set the completed square equal to \(-6\):
\[
(x - 3)^2 = -3
\]

4. Taking the square root of both sides:
\[
x - 3 = \pm \sqrt{-3}
\]

5. Solving for \(x\):
\[
x = 3 \pm \sqrt{3}i
\]

Now we can fill in the last part:
\[
x = 3 + \sqrt{3}i \quad \text{and} \quad x = 3 - \sqrt{3}i
\]

In summary, here’s your completed calculation:

- \( (x^2 - 6x + \underline{9}) + 6 - 9 = 0 \)
- \( (x - 3)^2 = -3 \)
- \( x - 3 = \underline{\pm \sqrt{-3}} \)
- \( x = 3 \pm \sqrt{3}i \)

Thus, the final answer is:
- \( x = 3 + \sqrt{3}i \) and \( x = 3 - \sqrt{3}i \)