Question 200505
{{{sqrt(-w)-6=w}}} Start with the given equation.



{{{sqrt(-w)=w+6}}} Add 6 to both sides.



{{{-w=(w+6)^2}}} Square both sides



{{{-w=w^2+12w+36}}} FOIL



{{{0=w^2+12w+36+w}}} Add w to both sides.



{{{0=w^2+13w+36}}} Combine like terms.



Notice that the quadratic {{{w^2+13w+36}}} is in the form of {{{Aw^2+Bw+C}}} where {{{A=1}}}, {{{B=13}}}, and {{{C=36}}}



Let's use the quadratic formula to solve for "w":



{{{w = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{w = (-(13) +- sqrt( (13)^2-4(1)(36) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=13}}}, and {{{C=36}}}



{{{w = (-13 +- sqrt( 169-4(1)(36) ))/(2(1))}}} Square {{{13}}} to get {{{169}}}. 



{{{w = (-13 +- sqrt( 169-144 ))/(2(1))}}} Multiply {{{4(1)(36)}}} to get {{{144}}}



{{{w = (-13 +- sqrt( 25 ))/(2(1))}}} Subtract {{{144}}} from {{{169}}} to get {{{25}}}



{{{w = (-13 +- sqrt( 25 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{w = (-13 +- 5)/(2)}}} Take the square root of {{{25}}} to get {{{5}}}. 



{{{w = (-13 + 5)/(2)}}} or {{{w = (-13 - 5)/(2)}}} Break up the expression. 



{{{w = (-8)/(2)}}} or {{{w =  (-18)/(2)}}} Combine like terms. 



{{{w = -4}}} or {{{w = -9}}} Simplify. 



So the <i>possible</i> solutions are {{{w = -4}}} or {{{w = -9}}} 



However, if you plug in {{{w = -9}}}, you get an inconsistent equation. 



So the only solution is {{{w = -4}}}