To determine which equation is equivalent to the given equation [tex]\(x^2 - 6x = 8\)[/tex], let's solve it step-by-step and convert it into the completed square form.
### Step 1: Move all terms to one side
First, we move 8 to the left side of the equation:
[tex]\[ x^2 - 6x - 8 = 0 \][/tex]
### Step 2: Complete the square
To complete the square, we follow these steps:
1. Start with the quadratic expression [tex]\(x^2 - 6x\)[/tex].
2. Take the coefficient of [tex]\(x\)[/tex], which is -6, halve it to get -3, and then square it to get 9.
3. Add and subtract this square (9) within the expression.
Here’s the process illustrated:
[tex]\[ x^2 - 6x = (x^2 - 6x + 9) - 9 \][/tex]
### Step 3: Rewrite the equation
Rewrite the expression as a perfect square minus the extra term:
[tex]\[ x^2 - 6x + 9 - 9 - 8 = 0 \][/tex]
[tex]\[ (x - 3)^2 - 17 = 0 \][/tex]
### Step 4: Simplify
Isolate the perfect square term by adding 17 to both sides:
[tex]\[ (x - 3)^2 = 17 \][/tex]
### Step 5: Compare with the options
We note that the completed square form of the given equation is:
[tex]\[ (x - 3)^2 = 17 \][/tex]
Now, let's match this with the provided options:
A. [tex]\((x - 6)^2 = 20\)[/tex] — This is incorrect.
B. [tex]\((x - 6)^2 = 44\)[/tex] — This is incorrect.
C. [tex]\((x - 3)^2 = 14\)[/tex] — This is incorrect.
D. [tex]\((x - 3)^2 = 17\)[/tex] — This is correct.
### Conclusion
Therefore, the correct answer is:
[tex]\[ \boxed{(D): (x - 3)^2 = 17} \][/tex]
