solution:
the given function is ,
![f(x) = 4 x^3 - 72 x^2 + 320 x,](/tex.php?f=f(x) = 4 x^3 - 72 x^2 + 320 x,)
= ![4x( x^2- 18 x + 80)\\\\ 4 x(x^2-10 x -8 x + 80) \\\\ 4 x[(x-10)(x-8)]](/tex.php?f=4x( x^2- 18 x + 80)\\\\ 4 x(x^2-10 x -8 x + 80) \\\\ 4 x[(x-10)(x-8)])
the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.
if a quadratic equation is of the form ,
sum of the roots = a + b +c= 0+8+10=18= ![\frac{-b}{a}](/tex.php?f=\frac{-b}{a})
product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 = ![\frac{c}{a}](/tex.php?f=\frac{c}{a})
product of roots = a b c = 0= ![\frac{d}{a}](/tex.php?f=\frac{d}{a})
there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function
is given by entry and exit route of this hall.