If the profit function for a firm is given by P(x)=-1600+170x-x^2 and limitations on space require that production be less than 100 units, find the break even points

To find the break-even points for the firm's profit function \( P(x) = -1600 + 170x - x^2 \), we need to determine the values of \( x \) where the profit is zero. In other words, we need to solve the equation:

\[
P(x) = 0
\]

\[
-1600 + 170x - x^2 = 0
\]

This is a quadratic equation. We can rewrite it as:

\[
-x^2 + 170x - 1600 = 0
\]

Now, we solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For our equation, \( a = -1 \), \( b = 170 \), and \( c = -1600 \). Substituting these values into the quadratic formula:

\[
x = \frac{-170 \pm \sqrt{170^2 - 4(-1)(-1600)}}{2(-1)}
\]

\[
x = \frac{-170 \pm \sqrt{28900 - 6400}}{-2}
\]

\[
x = \frac{-170 \pm \sqrt{22500}}{-2}
\]

\[
x =

If the profit function for a firm is given by P(x)=-1600+170x-x^2 and limitations on space require that production be less than 100 units, find the break-even points.

Find the break-even points for the quadratic profit function P(x)=-1600+170x-x^2. This solution uses the quadratic formula to solve for production levels where profit equals zero. Suitable for students in Grades 9-12.

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