Answer :
To convert the given standard form of the quadratic equation \( y = x^2 - 8x + 29 \) into its vertex form, follow these steps carefully:
### Step 1: Identifying the required transformation
The standard form of a quadratic equation is given by:
[tex]\[ y = ax^2 + bx + c \][/tex]
To convert this to the vertex form \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola, we need to complete the square.
### Step 2: Completing the square
Consider the quadratic and linear parts of the equation:
[tex]\[ x^2 - 8x \][/tex]
We will complete the square by following these steps:
1. Take half of the linear coefficient, \(-8\), and square it.
[tex]\[ \left( \frac{-8}{2} \right)^2 = (-4)^2 = 16 \][/tex]
2. Add and subtract the square inside the equation:
[tex]\[ x^2 - 8x + 16 - 16 \][/tex]
3. Reorganize the equation so that the perfect square trinomial is grouped together:
[tex]\[ (x^2 - 8x + 16) - 16 + 29 \][/tex]
4. Rewrite the perfect square trinomial as a square of a binomial:
[tex]\[ (x - 4)^2 - 16 + 29 \][/tex]
5. Combine constants to simplify:
[tex]\[ (x - 4)^2 + 13 \][/tex]
### Step 3: Writing the vertex form
After completing the square, we have transformed the equation into its vertex form:
[tex]\[ y = (x - 4)^2 + 13 \][/tex]
### Step 4: Selecting the correct option
Given the choices:
A. \( y = (x - 4)^2 + 13 \)
B. \( y = (x - 4)^2 + 28 \)
C. \( y = (x - 4)^2 + 18 \)
D. \( y = (x + 4)(x - 4) + 13 \)
The correct option that matches our derived vertex form is:
A. [tex]\( y = (x - 4)^2 + 13 \)[/tex]
### Step 1: Identifying the required transformation
The standard form of a quadratic equation is given by:
[tex]\[ y = ax^2 + bx + c \][/tex]
To convert this to the vertex form \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola, we need to complete the square.
### Step 2: Completing the square
Consider the quadratic and linear parts of the equation:
[tex]\[ x^2 - 8x \][/tex]
We will complete the square by following these steps:
1. Take half of the linear coefficient, \(-8\), and square it.
[tex]\[ \left( \frac{-8}{2} \right)^2 = (-4)^2 = 16 \][/tex]
2. Add and subtract the square inside the equation:
[tex]\[ x^2 - 8x + 16 - 16 \][/tex]
3. Reorganize the equation so that the perfect square trinomial is grouped together:
[tex]\[ (x^2 - 8x + 16) - 16 + 29 \][/tex]
4. Rewrite the perfect square trinomial as a square of a binomial:
[tex]\[ (x - 4)^2 - 16 + 29 \][/tex]
5. Combine constants to simplify:
[tex]\[ (x - 4)^2 + 13 \][/tex]
### Step 3: Writing the vertex form
After completing the square, we have transformed the equation into its vertex form:
[tex]\[ y = (x - 4)^2 + 13 \][/tex]
### Step 4: Selecting the correct option
Given the choices:
A. \( y = (x - 4)^2 + 13 \)
B. \( y = (x - 4)^2 + 28 \)
C. \( y = (x - 4)^2 + 18 \)
D. \( y = (x + 4)(x - 4) + 13 \)
The correct option that matches our derived vertex form is:
A. [tex]\( y = (x - 4)^2 + 13 \)[/tex]
