Question 156693
{{{4x + 2 = x^2}}} Start with the given equation



{{{4x + 2 - x^2=0}}} Subtract {{{x^2}}} from both sides



{{{- x^2+4x + 2 =0}}} Rearrange the terms.


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So let's complete the square for {{{-x^2+4x+2}}} 



{{{-x^2+4x+2}}} Start with the given expression.



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



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



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



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



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



{{{-1((x-2)^2-6)}}} Combine like terms.



{{{-1(x-2)^2-1(-6)}}} Distribute.



{{{-1(x-2)^2+6}}} Multiply.



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



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



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{{{-1(x-2)^2+6=0}}} Start with the given equation.



{{{-1(x-2)^2=0-6}}}Subtract {{{6}}} from both sides.



{{{-1(x-2)^2=-6}}} Combine like terms.



{{{(x-2)^2=(-6)/(-1)}}} Divide both sides by {{{-1}}}.



{{{(x-2)^2=6}}} Reduce.



{{{x-2=0+-sqrt(6)}}} Take the square root of both sides. Note: remember the "plus/minus"



{{{x-2=sqrt(6)}}} or {{{x-2=-sqrt(6)}}} Break up the "plus/minus" to form two equations.



{{{x=2+sqrt(6)}}} or {{{x=2-sqrt(6)}}} Add {{{2}}} to both sides.



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Answer:



So the solutions are {{{x=2+sqrt(6)}}} or {{{x=2-sqrt(6)}}}.