P(x)=3x-5/x+2

Solution for P(x)=3x-5/x+2 equation:

(P)=3P-5/P+2

We move all terms to the left:

(P)-(3P-5/P+2)=0


Domain of the equation: P+2)!=0

P∈R


We get rid of parentheses

P-3P+5/P-2=0

We multiply all the terms by the denominator


P*P-3P*P-2*P+5=0

We add all the numbers together, and all the variables

-2P+P*P-3P*P+5=0

Wy multiply elements

P^2-3P^2-2P+5=0

We add all the numbers together, and all the variables

-2P^2-2P+5=0

a = -2; b = -2; c = +5;
Δ = b2-4ac
Δ = -22-4·(-2)·5
Δ = 44
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:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{11}}{2*-2}=\frac{2-2\sqrt{11}}{-4}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{11}}{2*-2}=\frac{2+2\sqrt{11}}{-4}
$