To find an equation equivalent to [tex]\( x^2 - 6x = 8 \)[/tex], we will start by rearranging the expression and completing the square.
Step 1: Start with the given equation:
[tex]\[ x^2 - 6x = 8 \][/tex]
Step 2: Move the constant term to the left side to set the equation to zero:
[tex]\[ x^2 - 6x - 8 = 0 \][/tex]
Step 3: Complete the square on the left side.
[tex]\[ x^2 - 6x \][/tex]
To complete the square, we take half of the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-6\)[/tex]), then square it:
[tex]\[ \left(\frac{-6}{2}\right)^2 = 9 \][/tex]
Step 4: Add and subtract 9 within the equation to complete the square:
[tex]\[ x^2 - 6x + 9 - 9 - 8 = 0 \][/tex]
[tex]\[ (x - 3)^2 - 9 - 8 = 0 \][/tex]
Step 5: Simplify the equation:
[tex]\[ (x - 3)^2 - 17 = 0 \][/tex]
Step 6: Move 17 to the other side to isolate the square term:
[tex]\[ (x - 3)^2 = 17 \][/tex]
Therefore, the correct equation equivalent to the given equation [tex]\( x^2 - 6x = 8 \)[/tex] is:
[tex]\[ (x - 3)^2 = 17 \][/tex]
The correct answer is:
B. [tex]\((x - 3)^2 = 17\)[/tex]
