Question 167787


{{{x^2+6x+4}}} Start with the left side of the given equation.



Take half of the {{{x}}} coefficient {{{6}}} to get {{{3}}}. In other words, {{{(1/2)(6)=3}}}.



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



{{{x^2+6x+highlight(9-9)+4}}} Now add <font size=4><b>and</b></font> subtract {{{9}}}. Make sure to place this after the "x" term. Notice how {{{9-9=0}}}. So the expression is not changed.



{{{(x^2+6x+9)-9+4}}} Group the first three terms.



{{{(x+3)^2-9+4}}} Factor {{{x^2+6x+9}}} to get {{{(x+3)^2}}}.



{{{(x+3)^2-5}}} Combine like terms.



So after completing the square, {{{x^2+6x+4}}} transforms to {{{(x+3)^2-5}}}. So {{{x^2+6x+4=(x+3)^2-5}}}.



So {{{x^2+6x+4=0}}} is equivalent to {{{(x+3)^2-5=0}}}.




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{{{(x+3)^2-5=0}}} Start with the given equation.



{{{(x+3)^2=0+5}}}Add {{{5}}} to both sides.



{{{(x+3)^2=5}}} Combine like terms.



{{{x+3=0+-sqrt(5)}}} Take the square root of both sides.



{{{x+3=sqrt(5)}}} or {{{x+3=-sqrt(5)}}} Break up the "plus/minus" to form two equations.



{{{x=-3+sqrt(5)}}} or {{{x=-3-sqrt(5)}}} Subtract {{{3}}} from both sides.



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Answer:



So the solutions are {{{x=-3+sqrt(5)}}} or {{{x=-3-sqrt(5)}}}.