Question 170120


{{{z^2+8z+9}}} Start with the left side of the equation.



Take half of the {{{z}}} 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}}}



{{{z^2+8z+highlight(16-16)+9}}} Now add <font size=4><b>and</b></font> subtract {{{16}}}. Make sure to place this after the "z" term. Notice how {{{16-16=0}}}. So the expression is not changed.



{{{(z^2+8z+16)-16+9}}} Group the first three terms.



{{{(z+4)^2-16+9}}} Factor {{{z^2+8z+16}}} to get {{{(z+4)^2}}}.



{{{(z+4)^2-7}}} Combine like terms.



So after completing the square, {{{z^2+8z+9}}} transforms to {{{(z+4)^2-7}}}. So {{{z^2+8z+9=(z+4)^2-7}}}.



So {{{z^2+8z+9=0}}} is equivalent to {{{(z+4)^2-7=0}}}.



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{{{(z+4)^2-7=0}}} Start with the given equation.



{{{(z+4)^2=0+7}}}Add {{{7}}} to both sides.



{{{(z+4)^2=7}}} Combine like terms.



{{{x+4=0+-sqrt(7)}}} Take the square root of both sides.



{{{z+4=sqrt(7)}}} or {{{z+4=-sqrt(7)}}} Break up the "plus/minus" to form two equations.



{{{z+4=sqrt(7)}}} or {{{z+4=-sqrt(7)}}}  Simplify the square root.



{{{z=-4+sqrt(7)}}} or {{{z=-4-sqrt(7)}}} Subtract {{{4}}} from both sides.



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Answer:



So the solutions are {{{z=-4+sqrt(7)}}} or {{{z=-4-sqrt(7)}}}



which approximate to {{{z=-1.354}}} or {{{z=-6.646}}}