Question 195673
{{{3x^2-6x-24}}} Start with the left side of the given equation.



{{{3(x^2-2x-8)}}} 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)-8)}}} 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)-1-8)}}} Group the first three terms.



{{{3((x-1)^2-1-8)}}} Factor {{{x^2-2x+1}}} to get {{{(x-1)^2}}}.



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



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



{{{3(x-1)^2-27}}} Multiply.



So after completing the square, {{{3x^2-6x-24}}} transforms to {{{3(x-1)^2-27}}}. So {{{3x^2-6x-24=3(x-1)^2-27}}}.



So {{{3x^2-6x-24=0}}} is equivalent to {{{3(x-1)^2-27=0}}}.



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Now let's solve {{{3(x-1)^2-27=0}}}



{{{3(x-1)^2-27=0}}} Start with the given equation.



{{{3(x-1)^2=0+27}}}Add {{{27}}} to both sides.



{{{3(x-1)^2=27}}} Combine like terms.



{{{(x-1)^2=(27)/(3)}}} Divide both sides by {{{3}}}.



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



{{{x-1=0+-sqrt(9)}}} Take the square root of both sides.



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



{{{x-1=3}}} or {{{x-1=-3}}}  Take the square root of {{{9}}} to get {{{3}}}.



{{{x=1+3}}} or {{{x=1-3}}} Add {{{1}}} to both sides.



{{{x=4}}} or {{{x=-2}}} Combine like terms.



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Answer:



So the solutions are {{{x=4}}} or {{{x=-2}}}.