Please I need help to show how to solve P(x) = 0?

Here below my solution:

Our cost function is C(x) = 10x+100 and prive demand function is P(x) =35-x

now C(0) = 10*0 + 100 = 100

it means when no bottle is produced our fixed cost is $200

p(5) = 35-5 =30

it means when demand is of 5 bottles, price per bottle is $30.

C'(x) =10 so C'(10) = 10

it means marginal cost for producing 10th bottle is $10.

R(x)=revenue function= cost * demand

when demand is of x bottles then

R(x) =x. p(x)

= x(35-x)

=35x - x^2

now P(x) is profit function and it is

R(x) - C(x) = 35x-x2 - {10x+100}

= 35x - x2 - 10x - 100

= 25x - x2 - 100

when P(x) =0 we get

x2 - 25x +100 =0

solve and we get x=20 and 5

it means profit is maximum when demand is 5 bottles or 20 bottles