p=6+1/3p

Solution for p=6+1/3p equation:

p=6+1/3p

We move all terms to the left:

p-(6+1/3p)=0


Domain of the equation: 3p)!=0

p!=0/1

p!=0

p∈R


We add all the numbers together, and all the variables

p-(1/3p+6)=0

We get rid of parentheses

p-1/3p-6=0

We multiply all the terms by the denominator


p*3p-6*3p-1=0

Wy multiply elements

3p^2-18p-1=0

a = 3; b = -18; c = -1;
Δ = b2-4ac
Δ = -182-4·3·(-1)
Δ = 336
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:

$\sqrt{\Delta}=\sqrt{336}=\sqrt{16*21}=\sqrt{16}*\sqrt{21}=4\sqrt{21}$
$p_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-4\sqrt{21}}{2*3}=\frac{18-4\sqrt{21}}{6}
$
$p_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+4\sqrt{21}}{2*3}=\frac{18+4\sqrt{21}}{6}
$