Question 355000
First, let's complete the square for the expression {{{x^2-6x+3}}}



{{{x^2-6x+3}}} Start with the given expression.



Take half of the {{{x}}} coefficient {{{-6}}} to get {{{-3}}}. In other words, {{{(1/2)(-6)=-3}}}.



Now square {{{-3}}} to get {{{9}}}. In other words, {{{(-3)^2=(-3)(-3)=9}}}



{{{x^2-6x+highlight(9-9)+3}}} Now add <font size=4><b>and</b></font> subtract {{{9}}}. Make sure to place this after the "x" term. Notice how {{{9-9=0}}}. So the expression is not changed.



{{{(x^2-6x+9)-9+3}}} Group the first three terms.



{{{(x-3)^2-9+3}}} Factor {{{x^2-6x+9}}} to get {{{(x-3)^2}}}.



{{{(x-3)^2-6}}} Combine like terms.



So after completing the square, {{{x^2-6x+3}}} transforms to {{{(x-3)^2-6}}}. So {{{x^2-6x+3=(x-3)^2-6}}}.



So {{{x^2-6x+3=0}}} is equivalent to {{{(x-3)^2-6=0}}}.



Now let's solve {{{(x-3)^2-6=0}}}




{{{(x-3)^2-6=0}}} Start with the given equation.



{{{(x-3)^2=0+6}}}Add {{{6}}} to both sides.



{{{(x-3)^2=6}}} Combine like terms.



{{{x-3=""+-sqrt(6)}}} Take the square root of both sides.



{{{x-3=sqrt(6)}}} or {{{x-3=-sqrt(6)}}} Break up the "plus/minus" to form two equations.



{{{x=3+sqrt(6)}}} or {{{x=3-sqrt(6)}}} Add {{{3}}} to both sides.



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Answer:



So the solutions are {{{x=3+sqrt(6)}}} or {{{x=3-sqrt(6)}}}.



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