IDNLearn.com: Your go-to resource for finding precise and accurate answers. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
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].
### 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].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.