Question 189933


{{{x^2-4x+30}}} Start with the left side of the equation.



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}}}



{{{x^2-4x+highlight(4-4)+30}}} Now add <font size=4><b>and</b></font> subtract {{{4}}}. Make sure to place this after the "x" term. Notice how {{{4-4=0}}}. So the expression is not changed.



{{{(x^2-4x+4)-4+30}}} Group the first three terms.



{{{(x-2)^2-4+30}}} Factor {{{x^2-4x+4}}} to get {{{(x-2)^2}}}.



{{{(x-2)^2+26}}} Combine like terms.



So after completing the square, {{{x^2-4x+30}}} transforms to {{{(x-2)^2+26}}}. So {{{x^2-4x+30=(x-2)^2+26}}}.



So {{{x^2-4x+30=0}}} is equivalent to {{{(x-2)^2+26=0}}}.



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{{{(x-2)^2+26=0}}} Start with the given equation.



{{{(x-2)^2=0-26}}}Subtract {{{26}}} from both sides.



{{{(x-2)^2=-26}}} Combine like terms.



{{{x-2=0+-sqrt(-26)}}} Take the square root of both sides.



{{{x-2=sqrt(-26)}}} or {{{x-2=-sqrt(-26)}}} Break up the "plus/minus" to form two equations.



{{{x-2=i*sqrt(26)}}} or {{{x-2=-i*sqrt(26)}}}  Simplify the square root. Note: {{{i=sqrt(-1)}}}



{{{x=2+i*sqrt(26)}}} or {{{x=2-i*sqrt(26)}}} Add {{{2}}} to both sides.



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Answer:



So the solutions are {{{x=2+i*sqrt(26)}}} or {{{x=2-i*sqrt(26)}}}.