P(x)=4x^2-7x+1

Solution for P(x)=4x^2-7x+1 equation:

(P)=4P^2-7P+1

We move all terms to the left:

(P)-(4P^2-7P+1)=0

We get rid of parentheses

-4P^2+P+7P-1=0

We add all the numbers together, and all the variables

-4P^2+8P-1=0

a = -4; b = 8; c = -1;
Δ = b2-4ac
Δ = 82-4·(-4)·(-1)
Δ = 48
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-4\sqrt{3}}{2*-4}=\frac{-8-4\sqrt{3}}{-8}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+4\sqrt{3}}{2*-4}=\frac{-8+4\sqrt{3}}{-8}
$