Question 210196
{{{P = -5x^2 +1000x + 5000}}} Start with the given equation.



{{{55000 = -5x^2 +1000x + 5000}}} Plug in {{{P=55000}}}



{{{0= -5x^2 +1000x + 5000-55000 }}} Subtract 55000 from both sides.



{{{0= -5x^2 +1000x  -50000 }}} Combine like terms.



Notice that the quadratic {{{-5x^2+1000x-50000}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=-5}}}, {{{B=1000}}}, and {{{C=-50000}}}



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



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



{{{x = (-(1000) +- sqrt( (1000)^2-4(-5)(-50000) ))/(2(-5))}}} Plug in  {{{A=-5}}}, {{{B=1000}}}, and {{{C=-50000}}}



{{{x = (-1000 +- sqrt( 1000000-4(-5)(-50000) ))/(2(-5))}}} Square {{{1000}}} to get {{{1000000}}}. 



{{{x = (-1000 +- sqrt( 1000000-1000000 ))/(2(-5))}}} Multiply {{{4(-5)(-50000)}}} to get {{{1000000}}}



{{{x = (-1000 +- sqrt( 0 ))/(2(-5))}}} Subtract {{{1000000}}} from {{{1000000}}} to get {{{0}}}



{{{x = (-1000 +- sqrt( 0 ))/(-10)}}} Multiply {{{2}}} and {{{-5}}} to get {{{-10}}}. 



{{{x = (-1000 +- 0)/(-10)}}} Take the square root of {{{0}}} to get {{{0}}}. 



{{{x = (-1000 + 0)/(-10)}}} or {{{x = (-1000 - 0)/(-10)}}} Break up the expression. 



{{{x = (-1000)/(-10)}}} or {{{x =  (-1000)/(-10)}}} Combine like terms. 



{{{x = 100}}} or {{{x = 100}}} Simplify. 



So the only solution is {{{x = 100}}} 



This means that 100 computers must be sold to reach a profit of $55,000