To write the quadratic equation [tex]\( y = x^2 - 4x + 6 \)[/tex] in vertex form, which is [tex]\( y = a(x - h)^2 + k \)[/tex], follow these steps:
### Step 1: Identify coefficients
From the given quadratic equation [tex]\( y = x^2 - 4x + 6 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 6 \)[/tex]
### Step 2: Completing the square
To convert the quadratic equation to vertex form, we need to complete the square.
1. Start with the quadratic and linear terms inside the expression:
[tex]\[
y = x^2 - 4x + 6
\][/tex]
2. To complete the square, take half of the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-4\)[/tex]), square it, and then add and subtract this value inside the expression:
[tex]\[
\left( \frac{-4}{2} \right)^2 = (-2)^2 = 4
\][/tex]
3. Add and subtract this square within the original quadratic expression:
[tex]\[
y = x^2 - 4x + 4 - 4 + 6
\][/tex]
4. Group the perfect square trinomial and the constants:
[tex]\[
y = (x^2 - 4x + 4) + (6 - 4)
\][/tex]
5. Factor the perfect square trinomial and simplify:
[tex]\[
y = (x - 2)^2 + 2
\][/tex]
### Step 3: Write in vertex form
The equation [tex]\( y = (x - 2)^2 + 2 \)[/tex] is in vertex form where [tex]\( a = 1 \)[/tex], [tex]\( h = 2 \)[/tex], and [tex]\( k = 2 \)[/tex].
### Conclusion
Comparing this with the given options:
1. [tex]\( y = (x + 2)^2 - 4 \)[/tex]
2. [tex]\( y = (x + 2)^2 - 2 \)[/tex]
3. [tex]\( y = (x - 2)^2 + 2 \)[/tex]
4. [tex]\( y = (x - 2)^2 + 4 \)[/tex]
The correct vertex form of [tex]\( y = x^2 - 4x + 6 \)[/tex] is:
[tex]\[ y = (x - 2)^2 + 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{y = (x - 2)^2 + 2} \][/tex]
