Question

A student solved the equation x squared minus 6 x plus 6 equals 0 by completing the square.



Move numbers to the lines to complete the student's calculation.

Response area with 8 blank spaces
x squared minus 6 x plus 6 equals 0
left-parenthesis x squared minus 6 x plus
Blank space 1 empty
right-parenthesis plus
Blank space 2 empty
plus 6 equals 0

left-parenthesis x minus 3 right-parenthesis squared equals
Blank space 3 empty

x minus 3 equals plus or minus
Blank space 4 empty



x equals
Blank space 5 empty
plus
Blank space 6 empty
and x equals
Blank space 7 empty
plus
Blank space 8 empty

Answer options with 10 options.
-9
-6
-3
-√6
-√3
√3
√6
3
6
9

Answer 1

To complete the calculation of the equation \(x^2 - 6x + 6 = 0\) by completing the square, we follow these steps:

1. We start with the expression \(x^2 - 6x + 6 = 0\).
2. We need to complete the square on \(x^2 - 6x\). The term to complete the square would be \(\left(\frac{-6}{2}\right)^2 = 9\).
3. We rewrite the equation as \(x^2 - 6x + 9 - 9 + 6 = 0\), which simplifies to \((x - 3)^2 - 3 = 0\).
4. Rearranging gives us \((x - 3)^2 = 3\).
5. Taking the square root gives us \(x - 3 = \pm \sqrt{3}\).
6. Finally, solving for \(x\) gives \(x = 3 \pm \sqrt{3}\).

Now we can fill in the blanks:

1. Blank space 1: 9
2. Blank space 2: -9
3. Blank space 3: 3
4. Blank space 4: \(\sqrt{3}\)
5. Blank space 5: 3
6. Blank space 6: \(\sqrt{3}\)
7. Blank space 7: 3
8. Blank space 8: -\(\sqrt{3}\)

So the completed calculation looks like this:

\[
x^2 - 6x + 6 = 0
\]

\[
\left( x^2 - 6x + 9 \right) - 9 + 6 = 0
\]

\[
\left( x - 3 \right)^2 = 3
\]

\[
x - 3 = \pm \sqrt{3}
\]

\[
x = 3 + \sqrt{3} \quad \text{and} \quad x = 3 - \sqrt{3}
\]