Let's determine which of the given equations is equivalent to the original equation [tex]\(x^2 - 6x = 8\)[/tex].
First, we rewrite the given equation in a standard quadratic form:
[tex]\[ x^2 - 6x - 8 = 0 \][/tex]
Next, we consider the equivalence of this quadratic equation with each of the options provided.
### Option A: [tex]\((x - 6)^2 = 20\)[/tex]
Expand the left-hand side:
[tex]\[ (x - 6)^2 = x^2 - 12x + 36 \][/tex]
Substitute it back in the equation:
[tex]\[ x^2 - 12x + 36 = 20 \][/tex]
Subtract 20 from both sides to bring it to a standard form:
[tex]\[ x^2 - 12x + 16 = 0 \][/tex]
This standard form does not match [tex]\(x^2 - 6x - 8 = 0\)[/tex]. So, option A is not equivalent.
### Option B: [tex]\((x - 6)^2 = 44\)[/tex]
Expand the left-hand side:
[tex]\[ (x - 6)^2 = x^2 - 12x + 36 \][/tex]
Substitute it back in the equation:
[tex]\[ x^2 - 12x + 36 = 44 \][/tex]
Subtract 44 from both sides to bring it to a standard form:
[tex]\[ x^2 - 12x - 8 = 0 \][/tex]
This standard form does not match [tex]\(x^2 - 6x - 8 = 0\)[/tex]. So, option B is not equivalent.
### Option C: [tex]\((x - 3)^2 = 17\)[/tex]
Expand the left-hand side:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Substitute it back in the equation:
[tex]\[ x^2 - 6x + 9 = 17 \][/tex]
Subtract 17 from both sides to bring it to a standard form:
[tex]\[ x^2 - 6x - 8 = 0 \][/tex]
This matches the original equation [tex]\(x^2 - 6x - 8 = 0\)[/tex]. So, option C is equivalent.
### Option D: [tex]\((x - 3)^2 = 14\)[/tex]
Expand the left-hand side:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Substitute it back in the equation:
[tex]\[ x^2 - 6x + 9 = 14 \][/tex]
Subtract 14 from both sides to bring it to a standard form:
[tex]\[ x^2 - 6x - 5 = 0 \][/tex]
This standard form does not match [tex]\(x^2 - 6x - 8 = 0\)[/tex]. So, option D is not equivalent.
Thus, the equation equivalent to [tex]\(x^2 - 6x = 8\)[/tex] is:
[tex]\[ \boxed{(x - 3)^2 = 17} \][/tex]
So, the correct answer is option C.
