Question 166951


{{{x^2-x-1}}} Start with the given expression.



Take half of the {{{x}}} coefficient {{{-1}}} to get {{{-1/2}}}. In other words, {{{(1/2)(-1)=-1/2}}}.



Now square {{{-1/2}}} to get {{{1/4}}}. In other words, {{{(-1/2)^2=(-1/2)(-1/2)=1/4}}}



{{{x^2-x+highlight(1/4-1/4)-1}}} Now add <font size=4><b>and</b></font> subtract {{{1/4}}}. Make sure to place this after the "x" term. Notice how {{{1/4-1/4=0}}}. So the expression is not changed.



{{{(x^2-x+1/4)-1/4-1}}} Group the first three terms.



{{{(x-1/2)^2-1/4-1}}} Factor {{{x^2-x+1/4}}} to get {{{(x-1/2)^2}}}.



{{{(x-1/2)^2-5/4}}} Combine like terms.



So after completing the square, {{{x^2-x-1}}} transforms to {{{(x-1/2)^2-5/4}}}. So {{{x^2-x-1=(x-1/2)^2-5/4}}}.



So {{{x^2-x-1=0}}} is equivalent to {{{(x-1/2)^2-5/4=0}}}.




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{{{(x-1/2)^2-5/4=0}}} Start with the given equation.



{{{(x-1/2)^2=0+5/4}}}Add {{{5/4}}} to both sides.



{{{(x-1/2)^2=5/4}}} Combine like terms.



{{{x-1/2=0+-sqrt(5/4)}}} Take the square root of both sides.



{{{x-1/2=sqrt(5/4)}}} or {{{x-1/2=-sqrt(5/4)}}} Break up the "plus/minus" to form two equations.



{{{x-1/2=sqrt(5)/2}}} or {{{x-1/2=-sqrt(5)/2}}}  Simplify the square root.



{{{x=1/2+sqrt(5)/2}}} or {{{x=1/2-sqrt(5)/2}}} Add {{{1/2}}} to both sides.



{{{x=(1+sqrt(5))/(2)}}} or {{{x=(1-sqrt(5))/(2)}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(1+sqrt(5))/(2)}}} or {{{x=(1-sqrt(5))/(2)}}}.