Sure, let's rewrite the equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] by completing the square.
### Step-by-Step Solution:
1. Start with the equation:
[tex]\[ x^2 - 4x + 3 = 0 \][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[ x^2 - 4x = -3 \][/tex]
3. Complete the square on the left-hand side:
- Take half of the coefficient of [tex]\( x \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-4\)[/tex].
- Half of [tex]\(-4\)[/tex] is [tex]\(-2\)[/tex].
- Square [tex]\(-2\)[/tex] to get [tex]\(4\)[/tex].
4. Add this square to both sides of the equation:
[tex]\[ x^2 - 4x + 4 = -3 + 4 \][/tex]
5. The left-hand side is now a perfect square:
[tex]\[ (x - 2)^2 = 1 \][/tex]
So, the number you need to complete the square is [tex]\( -2 \)[/tex]. When written in the form [tex]\((x + d)^2\)[/tex], the equation becomes:
[tex]\[ (x - 2)^2 = 1 \][/tex]
Thus:
[tex]\[ (x + \square)^2 = (x - 2)^2 \][/tex]
The value of [tex]\( \square \)[/tex] is [tex]\(-2\)[/tex].
In conclusion, after completing the square, the equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] can be rewritten as:
[tex]\[ (x - 2)^2 = 1 \][/tex]
And the value of [tex]\( \square \)[/tex] is [tex]\( -2 \)[/tex].
