To complete the square and then factor the quadratic equation \( x^2 - 12x = 58 \), we follow these steps:
1. Move the constant term to the right side:
\[ x^2 - 12x - 58 = 0 \]
2. Add and subtract the square of half the coefficient of the linear term (12 in this case), which is \( (-12/2)^2 = 36 \), to both sides of the equation:
\[ x^2 - 12x + 36 - 36 - 58 = 0 \]
\[ (x - 6)^2 - 94 = 0 \]
3. Add 94 to both sides to isolate the square term:
\[ (x - 6)^2 = 94 \]
Now, to solve for x, we take the square root of both sides:
\[ x - 6 = \pm \sqrt{94} \]
Add 6 to both sides to get the two possible solutions for x:
\[ x = 6 \pm \sqrt{94} \]
So, the equation resulting from completing the square and factoring \( x^2 - 12x = 58 \) is:
\[ x = 6 \pm \sqrt{94} \]
