Question

Tanya runs a catering business. Based on her records, her weekly profit can be approximated by the function LaTeX: f\left(x\right)=2x^2-44x-150 f ( x ) = 2 x 2 − 44 x − 150 , where LaTeX: x x is the number of meals she caters. If LaTeX: f(x) f ( x ) is negative, it means that the business has lost money. What is the number of meals that Tanya needs to cater in order to break-even?

Answer 1

The function

$2x^2-44x-150$

is a parabola concave up, whose solutions are

$2x^2-44x-150=0 \iff x^2-22x-75=0$

from here, you can use the quadratic formula

$x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

to find that the solutions of the parabola are $-3,\ 25$

So, the parabola is positive if $x$ (which wouldn't make sense in our case) or $x$

So, if Tanya caters 25 meals she breaks even, and starting with the 26th meal she will begin to profit.