Answer
$x$ is between 2 and 11
Work Step by Step
$P(x) = -2x^2+26x-44$
Let $P(x)=0$
$-2x^2+26x-44=0$
$-2(x^2-13x+22)=0$
$-2(x-11)(x-2)=0$
$x-11=0$
$x-11+11=0$
$x=11$
$x-2=0$
$x-2+2=0$
$x=2$
Possible ranges: $(-∞,2)$, $(2,11)$, $(11,∞)$
Let $x=0$, $x=10$,$x=20$
$x=0$
$P(x) = -2x^2+26x-44$
$P(0) = -2*0^2+26*0-44$
$P(0) = -2*0+0-44$
$P(0) = 0-44$
$P(0)=-44$ (does not work since we need profit to be positive)
$x=10$
$P(x) = -2x^2+26x-44$
$P(10) = -2*10^2+26*10-44$
$P(10) = -2*100+260-44$
$P(10)=-200+216$
$P(10)= 16$ (works since profit is positive)
$x=20$
$P(x) = -2x^2+26x-44$
$P(20) = -2*20^2+26*20-44$
$P(20)=-2*400+520-44$
$P(20)=-800+520-44$
$P(20)=-324$ (doesn't work since this is not positive)
