Respuesta :
Solving quadratic equations, it is found that he needs to charge:
1. He needs to charge $40 to break even.
2. He needs to charge $30 for a profit of $600.
What is a quadratic function?
A quadratic function is given according to the following rule:
[tex]y = ax^2 + bx + c[/tex]
The solutions are:
- [tex]x_1 = \frac{-b + \sqrt{\Delta}}{2a}[/tex]
- [tex]x_2 = \frac{-b - \sqrt{\Delta}}{2a}[/tex]
In which:
[tex]\Delta = b^2 - 4ac[/tex]
The profit equation in this problem is:
P(x) = -3x² + 150x - 1200.
He breaks even when P(x) = 0, hence:
-3x² + 150x - 1200 = 0.
The coefficients are a = -3, b = 150, c = -1200, hence:
- [tex]\Delta = 150^2 - 4(-3)(-1200) = 8100[/tex]
- [tex]x_1 = \frac{-150 + \sqrt{8100}}{-6}[/tex]
- [tex]x_2 = \frac{-150 - \sqrt{8100}}{-6} = 40[/tex]
He needs to charge $40 to break even.
For a profit of $600, we have that P(x) = 600, hence:
-3x² + 150x - 1200 = 600.
-3x² + 150x - 1800 = 0.
The coefficients are a = -3, b = 150, c = -1800, hence:
- [tex]\Delta = 150^2 - 4(-3)(-1800) = 900[/tex]
- [tex]x_1 = \frac{-150 + \sqrt{900}}{-6}[/tex]
- [tex]x_2 = \frac{-150 - \sqrt{900}}{-6} = 30[/tex]
He needs to charge $30 for a profit of $600.
More can be learned about quadratic equations at https://brainly.com/question/24737967
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