Revenue is given by f(x) = -15x^2 - 30x + 200
Cost is given by y = 5x - 12
to break even, then Revenue = Cost
Then:
[tex]-15x^2-30x+200=5x-12[/tex]
Now, let's solve for x, which is the price
Step #1.
add 12 to both sides
[tex]-15x^2-30x+200+12=5x-12+12[/tex]
#2
Simplify
[tex]-15x^2-30x+212=5x[/tex]
#3
subtract 5x from both sides
[tex]-15x^2-30x+212-5x=5x-5x[/tex]
#4
Simplify
[tex]-15x^2-35x+212=0[/tex]
#5
Use the general formula to solve quadratic equation
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a};a^2x+bx+c=0[/tex][tex]x_{1,\: 2}=\frac{-\left(-35\right)\pm\sqrt{\left(-35\right)^2-4\left(-15\right)\cdot\:212}}{2\left(-15\right)}[/tex]
#6
[tex]x_1=\frac{-\left(-35\right)+\sqrt{13945}}{2\left(-15\right)},\: x_2=\frac{-\left(-35\right)-\sqrt{13945}}{2\left(-15\right)}[/tex]
#7
[tex]x=-\frac{35+\sqrt{13945}}{30},\: x=\frac{\sqrt{13945}-35}{30}[/tex]
#8
in decimal numbers
[tex]x=-5.10,\: x=2.77[/tex]
a negative value does not make sense in this case, therefore the answer is
d) $2.77