Question 198867
First, let's complete the square for {{{3x^2-2x-4}}}



{{{3x^2-2x-4}}} Start with the left side of the given equation.



{{{3(x^2-(2/3)x-4/3)}}} 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 {{{-2/3}}} to get {{{-1/3}}}. In other words, {{{(1/2)(-2/3)=-1/3}}}.



Now square {{{-1/3}}} to get {{{1/9}}}. In other words, {{{(-1/3)^2=(-1/3)(-1/3)=1/9}}}



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



{{{3((x^2-(2/3)x+1/9)-1/9-4/3)}}} Group the first three terms.



{{{3((x-1/3)^2-1/9-4/3)}}} Factor {{{x^2-(2/3)x+1/9}}} to get {{{(x-1/3)^2}}}.



{{{3((x-1/3)^2-13/9)}}} Combine like terms.



{{{3(x-1/3)^2+3(-13/9)}}} Distribute.



{{{3(x-1/3)^2-13/3}}} Multiply.



So after completing the square, {{{3x^2-2x-4}}} transforms to {{{3(x-1/3)^2-13/3}}}. So {{{3x^2-2x-4=3(x-1/3)^2-13/3}}}.



So {{{3x^2-2x-4=0}}} is equivalent to {{{3(x-1/3)^2-13/3=0}}}.


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Now let's solve {{{3(x-1/3)^2-13/3=0}}}




{{{3(x-1/3)^2-13/3=0}}} Start with the given equation.



{{{3(x-1/3)^2=0+13/3}}}Add {{{13/3}}} to both sides.



{{{3(x-1/3)^2=13/3}}} Combine like terms.



{{{(x-1/3)^2=(13/3)/(3)}}} Divide both sides by {{{3}}}.



{{{(x-1/3)^2=13/9}}} Reduce.



{{{x-1/3=""+-sqrt(13/9)}}} Take the square root of both sides.



{{{x-1/3=sqrt(13/9)}}} or {{{x-1/3=-sqrt(13/9)}}} Break up the "plus/minus" to form two equations.



{{{x-1/3=sqrt(13)/3}}} or {{{x-1/3=-sqrt(13)/3}}}  Simplify the square root.



{{{x=1/3+sqrt(13)/3}}} or {{{x=1/3-sqrt(13)/3}}} Add {{{1/3}}} to both sides.



{{{x=(1+sqrt(13))/(3)}}} or {{{x=(1-sqrt(13))/(3)}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(1+sqrt(13))/(3)}}} or {{{x=(1-sqrt(13))/(3)}}}.