Question 170121
{{{m^2=8m+3}}} Start with the given equation.



{{{m^2-8m-3=0}}} Get all terms to the left side.



Now let's complete the square:



{{{m^2-8m-3}}} Start with the left side of the equation.



Take half of the {{{m}}} 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}}}



{{{m^2-8m+highlight(16-16)-3}}} Now add <font size=4><b>and</b></font> subtract {{{16}}}. Make sure to place this after the "m" term. Notice how {{{16-16=0}}}. So the expression is not changed.



{{{(m^2-8m+16)-16-3}}} Group the first three terms.



{{{(m-4)^2-16-3}}} Factor {{{m^2-8m+16}}} to get {{{(m-4)^2}}}.



{{{(m-4)^2-19}}} Combine like terms.



So after completing the square, {{{m^2-8m-3}}} transforms to {{{(m-4)^2-19}}}. So {{{m^2-8m-3=(m-4)^2-19}}}.



So {{{m^2-8m-3=0}}} is equivalent to {{{(m-4)^2-19=0}}}.



-------------------------------------------------------------



{{{(m-4)^2-19=0}}} Start with the given equation.



{{{(m-4)^2=0+19}}}Add {{{19}}} to both sides.



{{{(m-4)^2=19}}} Combine like terms.



{{{x-4=0+-sqrt(19)}}} Take the square root of both sides.



{{{m-4=sqrt(19)}}} or {{{m-4=-sqrt(19)}}} Break up the "plus/minus" to form two equations.



{{{m=4+sqrt(19)}}} or {{{m=4-sqrt(19)}}} Add {{{4}}} to both sides.



--------------------------------------



Answer:



So the solutions are {{{m=4+sqrt(19)}}} or {{{m=4-sqrt(19)}}}



which approximate to {{{m=8.359}}} or {{{m=-0.359}}}