To complete the square and rewrite the quadratic equation [tex]\(y = x^2 - 6x + 14\)[/tex] in vertex form, follow these steps:
### Step 1: Identify the quadratic and linear coefficients
In the equation [tex]\(y = x^2 - 6x + 14\)[/tex], the quadratic coefficient (the coefficient of [tex]\(x^2\)[/tex]) is 1, and the linear coefficient (the coefficient of [tex]\(x\)[/tex]) is -6.
### Step 2: Half the linear coefficient and square it
Take the linear coefficient, -6, divide it by 2, and then square the result:
[tex]\[
\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
\][/tex]
### Step 3: Add and subtract this square inside the equation
Add and subtract 9 within the quadratic expression:
[tex]\[
y = x^2 - 6x + 9 - 9 + 14
\][/tex]
### Step 4: Rewrite the equation as a perfect square trinomial
Now, combine the first three terms into a perfect square trinomial, and combine the constants:
[tex]\[
y = (x^2 - 6x + 9) + (14 - 9)
\][/tex]
[tex]\[
y = (x - 3)^2 + 5
\][/tex]
### Step 5: Express the quadratic in vertex form
The equation is now in the vertex form:
[tex]\[
y = (x - 3)^2 + 5
\][/tex]
### Step 6: Identify the vertex
In the vertex form [tex]\(y = a(x - h)^2 + k\)[/tex], the vertex of the parabola is [tex]\((h, k)\)[/tex]. Here, [tex]\(h = 3\)[/tex] and [tex]\(k = 5\)[/tex]. Thus, the vertex is [tex]\((3, 5)\)[/tex].
### Step 7: Determine if it is a maximum or minimum point
The coefficient of the [tex]\((x - 3)^2\)[/tex] term is positive (1), indicating that the parabola opens upward. Therefore, the vertex represents a minimum point.
### Conclusion
The vertex of the quadratic equation [tex]\(y = x^2 - 6x + 14\)[/tex] is at [tex]\((3, 5)\)[/tex], and it represents a minimum point.
So, the correct answer is:
D. Minimum at [tex]\((3, 5)\)[/tex].
