p-3=1/2p+2

Solution for p-3=1/2p+2 equation:

p-3=1/2p+2

We move all terms to the left:

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


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

p∈R


We get rid of parentheses

p-1/2p-2-3=0

We multiply all the terms by the denominator


p*2p-2*2p-3*2p-1=0

Wy multiply elements

2p^2-4p-6p-1=0

We add all the numbers together, and all the variables

2p^2-10p-1=0

a = 2; b = -10; c = -1;
Δ = b2-4ac
Δ = -102-4·2·(-1)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$p_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*2}=\frac{10-6\sqrt{3}}{4}
$
$p_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*2}=\frac{10+6\sqrt{3}}{4}
$