Let's go through the steps to rewrite the given quadratic equation [tex]\(2x^2 - 12x + 1 = 0\)[/tex] in vertex form.
### Step-by-Step Solution:
#### Step 1: Factor out the coefficient of [tex]\(x^2\)[/tex]
Given equation:
[tex]\[2x^2 - 12x + 1 = 0\][/tex]
Factor out the 2 from the [tex]\(x\)[/tex]-terms:
[tex]\[2(x^2 - 6x) + 1 = 0\][/tex]
#### Step 2: Complete the square inside the parenthesis
To complete the square for the quadratic expression [tex]\(x^2 - 6x\)[/tex]:
1. Take half of the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-6\)[/tex]), square it, and add and subtract this value inside the parenthesis.
2. [tex]\[\left(\frac{-6}{2}\right)^2 = 9\][/tex]
So we add and subtract 9 inside the parenthesis:
[tex]\[2(x^2 - 6x + 9 - 9) + 1 = 0\][/tex]
This simplifies to:
[tex]\[2((x - 3)^2 - 9) + 1 = 0\][/tex]
#### Step 3: Move the constant out and combine
Distribute the 2 and simplify:
[tex]\[2(x - 3)^2 - 18 + 1 = 0\][/tex]
Here, collect all constant terms outside the squared term. Notice that:
[tex]\[2((x - 3)^2 - 9) + 1 - 18 = 0\][/tex]
Now we simplify further:
[tex]\[2(x - 3)^2 - 17 = 0\][/tex]
Thus, the correct form of the quadratic equation in vertex form is:
[tex]\[2(x - 3)^2 - 17 = 0\][/tex]
So, the missing step and the final correct vertex form of the equation is:
[tex]\[2((x - 3)^2 - 9) + 1 - 18 = 0\][/tex]
Given this, the correct vertex form of the equation is:
[tex]\[2(x - 3)^2 - 17 = 0\][/tex]
