Question 357118
{{{2x^2-x+6}}} Start with the given expression.



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



Now square {{{-1/4}}} to get {{{1/16}}}. In other words, {{{(-1/4)^2=(-1/4)(-1/4)=1/16}}}



{{{2(x^2-(1/2)x+highlight(1/16-1/16)+3)}}} Now add <font size=4><b>and</b></font> subtract {{{1/16}}} inside the parenthesis. Make sure to place this after the "x" term. Notice how {{{1/16-1/16=0}}}. So the expression is not changed.



{{{2((x^2-(1/2)x+1/16)-1/16+3)}}} Group the first three terms.



{{{2((x-1/4)^2-1/16+3)}}} Factor {{{x^2-(1/2)x+1/16}}} to get {{{(x-1/4)^2}}}.



{{{2((x-1/4)^2+47/16)}}} Combine like terms.



{{{2(x-1/4)^2+2(47/16)}}} Distribute.



{{{2(x-1/4)^2+47/8}}} Multiply.



So after completing the square, {{{2x^2-x+6}}} transforms to {{{2(x-1/4)^2+47/8}}}. So {{{2x^2-x+6=2(x-1/4)^2+47/8}}}.



So {{{2x^2-x+6=0}}} is equivalent to {{{2(x-1/4)^2+47/8=0}}}.



Now let's solve {{{2(x-1/4)^2+47/8=0}}}



{{{2(x-1/4)^2+47/8=0}}} Start with the given equation.



{{{2(x-1/4)^2=0-47/8}}}Subtract {{{47/8}}} from both sides.



{{{2(x-1/4)^2=-47/8}}} Combine like terms.



{{{(x-1/4)^2=(-47/8)/(2)}}} Divide both sides by {{{2}}}.



{{{(x-1/4)^2=-47/16}}} Reduce.



{{{x-1/4=""+-sqrt(-47/16)}}} Take the square root of both sides.



{{{x-1/4=sqrt(-47/16)}}} or {{{x-1/4=-sqrt(-47/16)}}} Break up the "plus/minus" to form two equations.



{{{x-1/4=i*(sqrt(47))/4)}}} or {{{x-1/4=-i*(sqrt(47))/4)}}}  Simplify the square root.



{{{x=1/4+i*(sqrt(47))/4)}}} or {{{x=1/4-i*(sqrt(47))/4)}}} Add {{{1/4}}} to both sides.



{{{x=(1+i*sqrt(47))/4}}} or {{{x=(1-i*sqrt(47))/4}}} Combine the fractions.



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Answer:



So the solutions are {{{x=(1+i*sqrt(47))/4}}} or {{{x=(1-i*sqrt(47))/4}}} where {{{i=sqrt(-1)}}}.



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