F(x)=x(1-x)-4x(x+2)+1

Solution for F(x)=x(1-x)-4x(x+2)+1 equation:

(F)=F(1-F)-4F(F+2)+1

We move all terms to the left:

(F)-(F(1-F)-4F(F+2)+1)=0

We add all the numbers together, and all the variables

F-(F(-1F+1)-4F(F+2)+1)=0


We calculate terms in parentheses: -(F(-1F+1)-4F(F+2)+1), so:

F(-1F+1)-4F(F+2)+1

We multiply parentheses

-1F^2-4F^2+F-8F+1

We add all the numbers together, and all the variables

-5F^2-7F+1

Back to the equation:

-(-5F^2-7F+1)


We get rid of parentheses

5F^2+7F+F-1=0

We add all the numbers together, and all the variables

5F^2+8F-1=0

a = 5; b = 8; c = -1;
Δ = b2-4ac
Δ = 82-4·5·(-1)
Δ = 84
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2\sqrt{21}}{2*5}=\frac{-8-2\sqrt{21}}{10}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2\sqrt{21}}{2*5}=\frac{-8+2\sqrt{21}}{10}
$