Question 198574
{{{3x^2-6x=27}}} Start with the given equation.



{{{3x^2-6x-27=0}}} Subtract 27 from both sides.



Now let's complete the square for the left side expression {{{3x^2-6x-27}}}



{{{3x^2-6x-27}}} Start with the given expression.



{{{3(x^2-2x-9)}}} Factor out the {{{x^2}}} coefficient {{{3}}}. This step is very important: the {{{x^2}}} coefficient <font size=4><b>must</b></font> be equal to 1.



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



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



{{{3(x^2-2x+highlight(1-1)-9)}}} Now add <font size=4><b>and</b></font> subtract {{{1}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{1-1=0}}}. So the expression is not changed.



{{{3((x^2-2x+1)-1-9)}}} Group the first three terms.



{{{3((x-1)^2-1-9)}}} Factor {{{x^2-2x+1}}} to get {{{(x-1)^2}}}.



{{{3((x-1)^2-10)}}} Combine like terms.



{{{3(x-1)^2+3(-10)}}} Distribute.



{{{3(x-1)^2-30}}} Multiply.



So after completing the square, {{{3x^2-6x-27}}} transforms to {{{3(x-1)^2-30}}}. So {{{3x^2-6x-27=3(x-1)^2-30}}}.



So {{{3x^2-6x-27=0}}} is equivalent to {{{3(x-1)^2-30=0}}}.


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Now let's solve {{{3(x-1)^2-30=0}}}



{{{3(x-1)^2-30=0}}} Start with the given equation.



{{{3(x-1)^2=0+30}}}Add {{{30}}} to both sides.



{{{3(x-1)^2=30}}} Combine like terms.



{{{(x-1)^2=(30)/(3)}}} Divide both sides by {{{3}}}.



{{{(x-1)^2=10}}} Reduce.



{{{x-1=""+-sqrt(10)}}} Take the square root of both sides.



{{{x-1=sqrt(10)}}} or {{{x-1=-sqrt(10)}}} Break up the "plus/minus" to form two equations.



{{{x-1=sqrt(10)}}} or {{{x-1=-sqrt(10)}}}  Simplify the square root.



{{{x=1+sqrt(10)}}} or {{{x=1-sqrt(10)}}} Add {{{1}}} to both sides.



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Answer:



So the solutions are {{{x=1+sqrt(10)}}} or {{{x=1-sqrt(10)}}}.




Note: you can use the quadratic formula to verify your answer.