Question 241751


{{{3x^2+14x+94}}} Start with the given expression.



{{{3(x^2+(14/3)x+94/3)}}} 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 {{{14/3}}} to get {{{7/3}}}. In other words, {{{(1/2)(14/3)=7/3}}}.



Now square {{{7/3}}} to get {{{49/9}}}. In other words, {{{(7/3)^2=(7/3)(7/3)=49/9}}}



{{{3(x^2+(14/3)x+highlight(49/9-49/9)+94/3)}}} Now add <font size=4><b>and</b></font> subtract {{{49/9}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{49/9-49/9=0}}}. So the expression is not changed.



{{{3((x^2+(14/3)x+49/9)-49/9+94/3)}}} Group the first three terms.



{{{3((x+7/3)^2-49/9+94/3)}}} Factor {{{x^2+(14/3)x+49/9}}} to get {{{(x+7/3)^2}}}.



{{{3((x+7/3)^2+233/9)}}} Combine like terms.



{{{3(x+7/3)^2+3(233/9)}}} Distribute.



{{{3(x+7/3)^2+233/3}}} Multiply.



So after completing the square, {{{3x^2+14x+94}}} transforms to {{{3(x+7/3)^2+233/3}}}. So {{{3x^2+14x+94=3(x+7/3)^2+233/3}}}.



So {{{3x^2+14x+94=0}}} is equivalent to {{{3(x+7/3)^2+233/3=0}}}.



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So to solve {{{3x^2+14x+94=0}}}, we can solve {{{3(x+7/3)^2+233/3=0}}}



{{{3(x+7/3)^2+233/3=0}}} Start with the given equation.



{{{3(x+7/3)^2=0-233/3}}}Subtract {{{233/3}}} from both sides.



{{{3(x+7/3)^2=-233/3}}} Combine like terms.



{{{(x+7/3)^2=(-233/3)/(3)}}} Divide both sides by {{{3}}}.



{{{(x+7/3)^2=-233/9}}} Reduce.



{{{x+7/3=""+-sqrt(-233/9)}}} Take the square root of both sides.



{{{x+7/3=sqrt(-233/9)}}} or {{{x+7/3=-sqrt(-233/9)}}} Break up the "plus/minus" to form two equations.



{{{x+7/3=(i*sqrt(233))/3}}} or {{{x+7/3=-(i*sqrt(233))/3}}}  Simplify the square root.



{{{x=-7/3+(i*sqrt(233))/3}}} or {{{x=-7/3-(i*sqrt(233))/3}}} Subtract {{{7/3}}} from both sides.



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Answer:



So the solutions are {{{x=-7/3+(i*sqrt(233))/3}}} or {{{x=-7/3-(i*sqrt(233))/3}}}.