3(1+2p)p=38+p

Solution for 3(1+2p)p=38+p equation:

Simplifying
3(1 + 2p) * p = 38 + p

Reorder the terms for easier multiplication:
3p(1 + 2p) = 38 + p
(1 * 3p + 2p * 3p) = 38 + p
(3p + 6p2) = 38 + p

Solving
3p + 6p2 = 38 + p

Solving for variable 'p'.

Reorder the terms:
-38 + 3p + -1p + 6p2 = 38 + p + -38 + -1p

Combine like terms: 3p + -1p = 2p
-38 + 2p + 6p2 = 38 + p + -38 + -1p

Reorder the terms:
-38 + 2p + 6p2 = 38 + -38 + p + -1p

Combine like terms: 38 + -38 = 0
-38 + 2p + 6p2 = 0 + p + -1p
-38 + 2p + 6p2 = p + -1p

Combine like terms: p + -1p = 0
-38 + 2p + 6p2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-19 + p + 3p2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-19 + p + 3p2)' equal to zero and attempt to solve:

Simplifying
-19 + p + 3p2 = 0

Solving
-19 + p + 3p2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-6.333333333 + 0.3333333333p + p2 = 0

Move the constant term to the right:

Add '6.333333333' to each side of the equation.
-6.333333333 + 0.3333333333p + 6.333333333 + p2 = 0 + 6.333333333

Reorder the terms:
-6.333333333 + 6.333333333 + 0.3333333333p + p2 = 0 + 6.333333333

Combine like terms: -6.333333333 + 6.333333333 = 0.000000000
0.000000000 + 0.3333333333p + p2 = 0 + 6.333333333
0.3333333333p + p2 = 0 + 6.333333333

Combine like terms: 0 + 6.333333333 = 6.333333333
0.3333333333p + p2 = 6.333333333

The p term is p.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333p + 0.25 + p2 = 6.333333333 + 0.25

Reorder the terms:
0.25 + 0.3333333333p + p2 = 6.333333333 + 0.25

Combine like terms: 6.333333333 + 0.25 = 6.583333333
0.25 + 0.3333333333p + p2 = 6.583333333

Factor a perfect square on the left side:
(p + 0.5)(p + 0.5) = 6.583333333

Calculate the square root of the right side: 2.56580072

Break this problem into two subproblems by setting 
(p + 0.5) equal to 2.56580072 and -2.56580072.

Subproblem 1
p + 0.5 = 2.56580072

Simplifying
p + 0.5 = 2.56580072

Reorder the terms:
0.5 + p = 2.56580072

Solving
0.5 + p = 2.56580072

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + p = 2.56580072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + p = 2.56580072 + -0.5
p = 2.56580072 + -0.5

Combine like terms: 2.56580072 + -0.5 = 2.06580072
p = 2.06580072

Simplifying
p = 2.06580072

Subproblem 2
p + 0.5 = -2.56580072

Simplifying
p + 0.5 = -2.56580072

Reorder the terms:
0.5 + p = -2.56580072

Solving
0.5 + p = -2.56580072

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + p = -2.56580072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + p = -2.56580072 + -0.5
p = -2.56580072 + -0.5

Combine like terms: -2.56580072 + -0.5 = -3.06580072
p = -3.06580072

Simplifying
p = -3.06580072

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {2.06580072, -3.06580072}
Solution
p = {2.06580072, -3.06580072}