f(X-1)=4X^2-5X+1

Solution for f(X-1)=4X^2-5X+1 equation:

f(f-1)=4f^2-5f+1

We move all terms to the left:

f(f-1)-(4f^2-5f+1)=0

We multiply parentheses

f^2-1f-(4f^2-5f+1)=0

We get rid of parentheses

f^2-4f^2-1f+5f-1=0

We add all the numbers together, and all the variables

-3f^2+4f-1=0

a = -3; b = 4; c = -1;
Δ = b2-4ac
Δ = 42-4·(-3)·(-1)
Δ = 4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{4}=2$
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2}{2*-3}=\frac{-6}{-6}
=1
$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2}{2*-3}=\frac{-2}{-6}
=1/3
$