To determine the number of units that give break even for the product, we need to find the values of [tex]\( x \)[/tex] that make the profit function [tex]\( P(x) \)[/tex] equal to zero. The profit function given is:
[tex]\[ P(x) = -11x^2 + 990x - 8800 \][/tex]
Setting the profit function equal to zero gives us the equation:
[tex]\[ -11x^2 + 990x - 8800 = 0 \][/tex]
This is a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = -11 \)[/tex], [tex]\( b = 990 \)[/tex], and [tex]\( c = -8800 \)[/tex].
To solve the quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the values [tex]\( a = -11 \)[/tex], [tex]\( b = 990 \)[/tex], and [tex]\( c = -8800 \)[/tex] into the formula:
[tex]\[ x = \frac{-990 \pm \sqrt{990^2 - 4(-11)(-8800)}}{2(-11)} \][/tex]
[tex]\[ x = \frac{-990 \pm \sqrt{980100 - 387200}}{-22} \][/tex]
[tex]\[ x = \frac{-990 \pm \sqrt{592900}}{-22} \][/tex]
[tex]\[ x = \frac{-990 \pm 770}{-22} \][/tex]
Now, we have two possible solutions:
1) [tex]\( x = \frac{-990 + 770}{-22} \)[/tex]
[tex]\[ x = \frac{-220}{-22} \][/tex]
[tex]\[ x = 10 \][/tex]
2) [tex]\( x = \frac{-990 - 770}{-22} \)[/tex]
[tex]\[ x = \frac{-1760}{-22} \][/tex]
[tex]\[ x = 80 \][/tex]
Thus, the number of units that give break even for the product are [tex]\( 10 \)[/tex] and [tex]\( 80 \)[/tex].
So, the number of units that break even for this product are [tex]\( \boxed{10, 80} \)[/tex].
