A(x)=x(50-x)

Solution for A(x)=x(50-x) equation:

(A)=A(50-A)

We move all terms to the left:

(A)-(A(50-A))=0

We add all the numbers together, and all the variables

A-(A(-1A+50))=0


We calculate terms in parentheses: -(A(-1A+50)), so:

A(-1A+50)

We multiply parentheses

-1A^2+50A

Back to the equation:

-(-1A^2+50A)


We get rid of parentheses

1A^2-50A+A=0

We add all the numbers together, and all the variables

A^2-49A=0

a = 1; b = -49; c = 0;
Δ = b2-4ac
Δ = -492-4·1·0
Δ = 2401
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{2401}=49$
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-49)-49}{2*1}=\frac{0}{2}
=0
$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-49)+49}{2*1}=\frac{98}{2}
=49
$