(5/3)p-(8/3)=2p+1

Solution for (5/3)p-(8/3)=2p+1 equation:

(5/3)p-(8/3)=2p+1

We move all terms to the left:

(5/3)p-(8/3)-(2p+1)=0


Domain of the equation: 3)p!=0

p!=0/1

p!=0

p∈R


We add all the numbers together, and all the variables

(+5/3)p-(2p+1)-(+8/3)=0

We multiply parentheses

5p^2-(2p+1)-(+8/3)=0

We get rid of parentheses

5p^2-2p-1-8/3=0

We multiply all the terms by the denominator


5p^2*3-2p*3-8-1*3=0

We add all the numbers together, and all the variables

5p^2*3-2p*3-11=0

Wy multiply elements

15p^2-6p-11=0

a = 15; b = -6; c = -11;
Δ = b2-4ac
Δ = -62-4·15·(-11)
Δ = 696
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:

$\sqrt{\Delta}=\sqrt{696}=\sqrt{4*174}=\sqrt{4}*\sqrt{174}=2\sqrt{174}$
$p_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{174}}{2*15}=\frac{6-2\sqrt{174}}{30}
$
$p_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{174}}{2*15}=\frac{6+2\sqrt{174}}{30}
$