Linear-Quadratic Systems

Question

Skylar repairs microwaves. Her revenue, in dollars, can be modeled by the equation y = 80 + 30x, where x is the number of hours spent repairing microwaves. Her overhead cost, in dollars, can be modeled by the equation y=10x2−20 , where xis the number of hours spent repairing microwaves.

After how many hours does she break even?

Note: The phrase break even refers to the value where the two functions are equivalent.

Answer 1

As the note says, break even refers to the value of "x" when the two functions are equal.

So,

80+30x=10x^2-20

To solve a quadratic equation, you need to bring all the terms on the one side of the equation to set the equation equal to zero.

10x^2-30x-100=0

Now solve by factoring:

10(x+2)(x-5)=0

x+2=0 or x-5=0

x=-2 or 5.

In this context, the answer of -2 makes no sense. So, she will break even when she spends 5 hours repairing microwaves.