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.
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.
Answers
GPT-4o mini
Let's complete the student's calculation step by step, filling in the blanks as needed.
Starting with the equation:
\[ x^2 - 6x + 6 = 0 \]
We can complete the square, which typically involves taking half of the coefficient of \( x \), squaring it, and then adding and subtracting it. The coefficient of \( x \) is -6, so:
1. Half of -6 is -3, and squaring it gives us \( 9 \).
2. We can rewrite the equation as:
\[ \left( x^2 - 6x + 9 \right) - 9 + 6 = 0 \]
This means we can write:
\[ \left( x^2 - 6x + 9 \right) - 3 = 0 \]
Now, we can write out the completed square:
\[ \left( x - 3 \right)^2 - 3 = 0 \]
Now, let's fill in the blanks in the given response area:
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 placed values will look like this:
- The completed statement will read upon filling:
\[ x^2 - 6x + 6 = 0 \]
\[ \left( x^2 - 6x + 9 \right) - 9 + 6 = 0 \]
\[ \left( x - 3 \right)^2 = 3 \]
\[ x - 3 = \pm \sqrt{3} \]
\[ x = 3 + \sqrt{3} \]
and
\[ x = 3 - \sqrt{3} \]
Let me know if you need anything else!
Starting with the equation:
\[ x^2 - 6x + 6 = 0 \]
We can complete the square, which typically involves taking half of the coefficient of \( x \), squaring it, and then adding and subtracting it. The coefficient of \( x \) is -6, so:
1. Half of -6 is -3, and squaring it gives us \( 9 \).
2. We can rewrite the equation as:
\[ \left( x^2 - 6x + 9 \right) - 9 + 6 = 0 \]
This means we can write:
\[ \left( x^2 - 6x + 9 \right) - 3 = 0 \]
Now, we can write out the completed square:
\[ \left( x - 3 \right)^2 - 3 = 0 \]
Now, let's fill in the blanks in the given response area:
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 placed values will look like this:
- The completed statement will read upon filling:
\[ x^2 - 6x + 6 = 0 \]
\[ \left( x^2 - 6x + 9 \right) - 9 + 6 = 0 \]
\[ \left( x - 3 \right)^2 = 3 \]
\[ x - 3 = \pm \sqrt{3} \]
\[ x = 3 + \sqrt{3} \]
and
\[ x = 3 - \sqrt{3} \]
Let me know if you need anything else!