To convert the quadratic equation from its standard form [tex]\( y = x^2 - 6x + 14 \)[/tex] to its vertex form, we need to complete the square. Let's go through this step-by-step:
1. Start with the given standard form:
[tex]\[
y = x^2 - 6x + 14
\][/tex]
2. Isolate the [tex]\( x \)[/tex] terms:
[tex]\[
y - 14 = x^2 - 6x
\][/tex]
3. Complete the square on the right-hand side:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-6\)[/tex], divide it by 2, and then square it.
[tex]\[
\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
\][/tex]
- Add and subtract this square inside the equation to maintain equality.
[tex]\[
y - 14 + 9 = x^2 - 6x + 9
\][/tex]
4. Simplify the equation:
- Combine constants on the left-hand side:
[tex]\[
y - 5 = x^2 - 6x + 9
\][/tex]
- Recognize that [tex]\( x^2 - 6x + 9 \)[/tex] is a perfect square trinomial, which can be expressed as [tex]\( (x - 3)^2 \)[/tex]:
[tex]\[
y - 5 = (x - 3)^2
\][/tex]
5. Isolate [tex]\( y \)[/tex] to write in vertex form:
[tex]\[
y = (x - 3)^2 + 5
\][/tex]
So, the vertex form of the given quadratic equation [tex]\( y = x^2 - 6x + 14 \)[/tex] is:
[tex]\[
y = (x - 3)^2 + 5
\][/tex]
Comparing this with the given choices, the correct answer is:
A. [tex]\( y = (x - 3)^2 + 5 \)[/tex]
