The profit for a product is given by $\newline$ $P(x)=-11x^{2}+770x-6600$ , where $\newline$ $x$ is the number of units produced and sold. How many units give break even (that is, give zero profit) for this product? $\newline$ The number of units that give the break even for this product are $\newline$ $\boxed{\phantom{000}}$ . $\newline$ (Use a comma to separate answers as needed.)

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Q. The profit for a product is given by

\newline

P(x)=-11x^{2}+770x-6600

, where

\newline

x

is the number of units produced and sold. How many units give break even (that is, give zero profit) for this product?

\newline

The number of units that give the break even for this product are

\newline

\boxed{\phantom{000}}

\newline

(Use a comma to separate answers as needed.)

Set Profit Function Equal: Set the profit function equal to zero to find the break-even points. $\newline$ $P(x) = -11x^2 + 770x - 6600$ $\newline$ To find the break-even points, we need to solve for $x$ when $P(x) = 0$ . $\newline$ $0 = -11x^2 + 770x - 6600$
Factor or Use Formula: Factor the quadratic equation or use the quadratic formula to find the values of $x$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$ . In our case, $a = -11$ , $b = 770$ , and $c = -6600$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ to ensure that the roots are real numbers. $\newline$ Discriminant = $b^2 - 4ac = (770)^2 - 4(-11)(-6600)$ $\newline$ Discriminant = $592900 - 290400$ $\newline$ Discriminant = $302500$ $\newline$ Since the discriminant is positive, we have two real roots.
Apply Quadratic Formula: Apply the quadratic formula to find the roots. $\newline$ $x = \frac{-770 \pm \sqrt{302500}}{2 \times -11}$ $\newline$ $x = \frac{-770 \pm 550}{-22}$
Solve for Values: Solve for the two values of $x$ .
$x_1 = \frac{{-770 + 550}}{{-22}}$
$x_1 = \frac{{-220}}{{-22}}$
$x_1 = 10$

$x2 = \frac{{-770 - 550}}{{-22}}$
$x2 = \frac{{-1320}}{{-22}}$
$x2 = 60$
Verify Valid Solutions: Verify that the roots make sense in the context of the problem. $\newline$ Since we are looking for the number of units produced and sold, negative values do not make sense. Both $10$ and $60$ are positive, so they are valid solutions for the number of units that give break-even.