Question 189904
{{{y^2 - 12y = -11}}} Start with the given equation.



{{{y^2 - 12y +11=0}}} Add 11 to both sides.



{{{y^2-12y+11}}} Start with the left side of the equation.



Take half of the {{{y}}} coefficient {{{-12}}} to get {{{-6}}}. In other words, {{{(1/2)(-12)=-6}}}.



Now square {{{-6}}} to get {{{36}}}. In other words, {{{(-6)^2=(-6)(-6)=36}}}



{{{y^2-12y+highlight(36-36)+11}}} Now add <font size=4><b>and</b></font> subtract {{{36}}}. Make sure to place this after the "y" term. Notice how {{{36-36=0}}}. So the expression is not changed.



{{{(y^2-12y+36)-36+11}}} Group the first three terms.



{{{(y-6)^2-36+11}}} Factor {{{y^2-12y+36}}} to get {{{(y-6)^2}}}.



{{{(y-6)^2-25}}} Combine like terms.



So after completing the square, {{{y^2-12y+11}}} transforms to {{{(y-6)^2-25}}}. So {{{y^2-12y+11=(y-6)^2-25}}}.



So {{{y^2-12y+11=0}}} is equivalent to {{{(y-6)^2-25=0}}}.



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{{{(y-6)^2-25=0}}} Start with the given equation.



{{{(y-6)^2=0+25}}}Add {{{25}}} to both sides.



{{{(y-6)^2=25}}} Combine like terms.



{{{x-6=""+-sqrt(25)}}} Take the square root of both sides.



{{{y-6=sqrt(25)}}} or {{{y-6=-sqrt(25)}}} Break up the "plus/minus" to form two equations.



{{{y-6=5}}} or {{{y-6=-5}}}  Take the square root of {{{25}}} to get {{{5}}}.



{{{y=6+5}}} or {{{y=6-5}}} Add {{{6}}} to both sides.



{{{y=11}}} or {{{y=1}}} Combine like terms.



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Answer:



So the solutions are {{{y=11}}} or {{{y=1}}}.