A(x)=x(500-2x)

Solution for A(x)=x(500-2x) equation:

(A)=A(500-2A)

We move all terms to the left:

(A)-(A(500-2A))=0

We add all the numbers together, and all the variables

A-(A(-2A+500))=0


We calculate terms in parentheses: -(A(-2A+500)), so:

A(-2A+500)

We multiply parentheses

-2A^2+500A

Back to the equation:

-(-2A^2+500A)


We get rid of parentheses

2A^2-500A+A=0

We add all the numbers together, and all the variables

2A^2-499A=0

a = 2; b = -499; c = 0;
Δ = b2-4ac
Δ = -4992-4·2·0
Δ = 249001
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{249001}=499$
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-499)-499}{2*2}=\frac{0}{4}
=0
$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-499)+499}{2*2}=\frac{998}{4}
=249+1/2
$