Question 201572
{{{n^2+16n-97=-3}}} Start with the given equation.



{{{n^2+16n-97+3=0}}} Add 3 to both sides.



{{{n^2+16n-94=0}}} Combine like terms.


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Now let's complete the square for the expression {{{n^2+16n-94}}}



{{{n^2+16n-94}}} Start with the given expression.



Take half of the {{{n}}} coefficient {{{16}}} to get {{{8}}}. In other words, {{{(1/2)(16)=8}}}.



Now square {{{8}}} to get {{{64}}}. In other words, {{{(8)^2=(8)(8)=64}}}



{{{n^2+16n+highlight(64-64)-94}}} Now add <font size=4><b>and</b></font> subtract {{{64}}}. Make sure to place this after the "n" term. Notice how {{{64-64=0}}}. So the expression is not changed.



{{{(n^2+16n+64)-64-94}}} Group the first three terms.



{{{(n+8)^2-64-94}}} Factor {{{n^2+16n+64}}} to get {{{(n+8)^2}}}.



{{{(n+8)^2-158}}} Combine like terms.



So after completing the square, {{{n^2+16n-94}}} transforms to {{{(n+8)^2-158}}}. So {{{n^2+16n-94=(n+8)^2-158}}}.



So {{{n^2+16n-94=0}}} is equivalent to {{{(n+8)^2-158=0}}}.



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Now let's solve {{{n^2+16n-94=0}}}



{{{(n+8)^2-158=0}}} Start with the given equation.



{{{(n+8)^2=0+158}}} Add {{{158}}} to both sides.



{{{(n+8)^2=158}}} Combine like terms.



{{{x+8=""+-sqrt(158)}}} Take the square root of both sides.



{{{n+8=sqrt(158)}}} or {{{n+8=-sqrt(158)}}} Break up the "plus/minus" to form two equations.



{{{n=-8+sqrt(158)}}} or {{{n=-8-sqrt(158)}}} Subtract {{{8}}} from both sides.



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Answer:



So the solutions are {{{n=-8+sqrt(158)}}} or {{{n=-8-sqrt(158)}}}.