P2+3=p(2p+1)+1

Solution for P2+3=p(2p+1)+1 equation:

2+3=P(2P+1)+1

We move all terms to the left:

2+3-(P(2P+1)+1)=0

We add all the numbers together, and all the variables

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


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

P(2P+1)+1

We multiply parentheses

2P^2+P+1

Back to the equation:

-(2P^2+P+1)


We get rid of parentheses

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

We add all the numbers together, and all the variables

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

a = -2; b = -1; c = +4;
Δ = b2-4ac
Δ = -12-4·(-2)·4
Δ = 33
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-\sqrt{33}}{2*-2}=\frac{1-\sqrt{33}}{-4}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+\sqrt{33}}{2*-2}=\frac{1+\sqrt{33}}{-4}
$