A(w)=w(84-2w)

Solution for A(w)=w(84-2w) equation:

(A)=A(84-2A)

We move all terms to the left:

(A)-(A(84-2A))=0

We add all the numbers together, and all the variables

A-(A(-2A+84))=0


We calculate terms in parentheses: -(A(-2A+84)), so:

A(-2A+84)

We multiply parentheses

-2A^2+84A

Back to the equation:

-(-2A^2+84A)


We get rid of parentheses

2A^2-84A+A=0

We add all the numbers together, and all the variables

2A^2-83A=0

a = 2; b = -83; c = 0;
Δ = b2-4ac
Δ = -832-4·2·0
Δ = 6889
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{6889}=83$
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-83)-83}{2*2}=\frac{0}{4}
=0
$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-83)+83}{2*2}=\frac{166}{4}
=41+1/2
$