Question 153209
Remember, 


Profit =  Revenue - Cost


which is symbolically:



{{{P(x)=R(x)-C(x)}}}



{{{P(x)=(56x - 2x^2)-(24x + 101)}}} Plug in {{{R(x) = 56x - 2x^2}}} and {{{C(x) = 24x + 101 }}}



{{{P(x)=56x - 2x^2-24x - 101}}} Distribute the negative.



{{{P(x)=-2x^2+32x-101}}} Combine like terms



Now the break even point occurs when you neither gain money nor lose money. In other words, break even point happens when the profit is zero. So this means that the break even point occurs when {{{P(x)=0}}}



{{{0=-2x^2+32x-101}}} Plug in {{{P(x)=0}}}



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=-2}}}, {{{b=32}}}, and {{{c=-101}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(32) +- sqrt( (32)^2-4(-2)(-101) ))/(2(-2))}}} Plug in  {{{a=-2}}}, {{{b=32}}}, and {{{c=-101}}}



{{{x = (-32 +- sqrt( 1024-4(-2)(-101) ))/(2(-2))}}} Square {{{32}}} to get {{{1024}}}. 



{{{x = (-32 +- sqrt( 1024-808 ))/(2(-2))}}} Multiply {{{4(-2)(-101)}}} to get {{{808}}}



{{{x = (-32 +- sqrt( 216 ))/(2(-2))}}} Subtract {{{808}}} from {{{1024}}} to get {{{216}}}



{{{x = (-32 +- sqrt( 216 ))/(-4)}}} Multiply {{{2}}} and {{{-2}}} to get {{{-4}}}. 



{{{x = (-32 +- 6*sqrt(6))/(-4)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (-32+6*sqrt(6))/(-4)}}} or {{{x = (-32-6*sqrt(6))/(-4)}}} Break up the expression.  



So the answers are {{{x = (-32+6*sqrt(6))/(-4)}}} or {{{x = (-32-6*sqrt(6))/(-4)}}} 



which approximate to {{{x=4.33}}} or {{{x=11.67}}} 



So the break even points occur when the price is either $4.33 or $11.67