Question 173222


{{{x^2+8x-11}}} Start with the given left side of the given equation.



Take half of the {{{x}}} coefficient {{{8}}} to get {{{4}}}. In other words, {{{(1/2)(8)=4}}}.



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



{{{x^2+8x+highlight(16-16)-11}}} Now add <font size=4><b>and</b></font> subtract {{{16}}}. Make sure to place this after the "x" term. Notice how {{{16-16=0}}}. So the expression is not changed.



{{{(x^2+8x+16)-16-11}}} Group the first three terms.



{{{(x+4)^2-16-11}}} Factor {{{x^2+8x+16}}} to get {{{(x+4)^2}}}.



{{{(x+4)^2-27}}} Combine like terms.



So after completing the square, {{{x^2+8x-11}}} transforms to {{{(x+4)^2-27}}}. So {{{x^2+8x-11=(x+4)^2-27}}}.



So {{{x^2+8x-11=0}}} is equivalent to {{{(x+4)^2-27=0}}}.



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{{{(x+4)^2-27=0}}} Start with the given equation.



{{{(x+4)^2=0+27}}}Add {{{27}}} to both sides.



{{{(x+4)^2=27}}} Combine like terms.



{{{x+4=0+-sqrt(27)}}} Take the square root of both sides.



{{{x+4=sqrt(27)}}} or {{{x+4=-sqrt(27)}}} Break up the "plus/minus" to form two equations.



{{{x+4=3*sqrt(3)}}} or {{{x+4=-3*sqrt(3)}}}  Simplify the square root.



{{{x=-4+3*sqrt(3)}}} or {{{x=-4-3*sqrt(3)}}} Subtract {{{4}}} from both sides.



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Answer:



So the solutions are {{{x=-4+3*sqrt(3)}}} or {{{x=-4-3*sqrt(3)}}}.