Question 487112
Your mistake is in steps 2 and 3.  Review the "completing the square" section in your textbook.
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{{{x^2 - 2x - 13 = 0 }}}
{{{x^2 - 2x = 13 }}}
Take 1/2 of the coefficient of the x term, square it, and add to both sides:
{{{x^2 - 2x + 1 = 13 + 1 }}}
{{{x^2 - 2x + 1 = 14 }}}
factor left:
{{{(x-1)(x-1) = 14 }}}
{{{(x-1)^2 = 14 }}}
{{{ x-1 = sqrt(14) }}}
{{{ x = sqrt(14) + 1 }}}