A(x)=2x(5-x)-(5-x)x-1

Solution for A(x)=2x(5-x)-(5-x)x-1 equation:

(A)=2A(5-A)-(5-A)A-1

We move all terms to the left:

(A)-(2A(5-A)-(5-A)A-1)=0

We add all the numbers together, and all the variables

A-(2A(-1A+5)-(-1A+5)A-1)=0


We calculate terms in parentheses: -(2A(-1A+5)-(-1A+5)A-1), so:

2A(-1A+5)-(-1A+5)A-1

We multiply parentheses

-2A^2+1A^2+10A-5A-1

We add all the numbers together, and all the variables

-1A^2+5A-1

Back to the equation:

-(-1A^2+5A-1)


We get rid of parentheses

1A^2-5A+A+1=0

We add all the numbers together, and all the variables

A^2-4A+1=0

a = 1; b = -4; c = +1;
Δ = b2-4ac
Δ = -42-4·1·1
Δ = 12
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{12}=\sqrt{4*3}=\sqrt{4}*\sqrt{3}=2\sqrt{3}$
$A_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{3}}{2*1}=\frac{4-2\sqrt{3}}{2}
$
$A_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{3}}{2*1}=\frac{4+2\sqrt{3}}{2}
$