24p^2+126p+108=0

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Solution for 24p^2+126p+108=0 equation: 


Simplifying
24p2 + 126p + 108 = 0

Reorder the terms:
108 + 126p + 24p2 = 0

Solving
108 + 126p + 24p2 = 0

Solving for variable 'p'.

Factor out the Greatest Common Factor (GCF), '6'.
6(18 + 21p + 4p2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(18 + 21p + 4p2)' equal to zero and attempt to solve:

Simplifying
18 + 21p + 4p2 = 0

Solving
18 + 21p + 4p2 = 0

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

Divide each side by '4'.
4.5 + 5.25p + p2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 5.25p + -4.5 + p2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 5.25p + p2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 5.25p + p2 = 0 + -4.5
5.25p + p2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
5.25p + p2 = -4.5

The p term is 5.25p.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25p + 6.890625 + p2 = -4.5 + 6.890625

Reorder the terms:
6.890625 + 5.25p + p2 = -4.5 + 6.890625

Combine like terms: -4.5 + 6.890625 = 2.390625
6.890625 + 5.25p + p2 = 2.390625

Factor a perfect square on the left side:
(p + 2.625)(p + 2.625) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(p + 2.625) equal to 1.54616461 and -1.54616461.


Subproblem 1
p + 2.625 = 1.54616461

Simplifying
p + 2.625 = 1.54616461

Reorder the terms:
2.625 + p = 1.54616461

Solving
2.625 + p = 1.54616461

Solving for variable 'p'.

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

Add '-2.625' to each side of the equation.
2.625 + -2.625 + p = 1.54616461 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + p = 1.54616461 + -2.625
p = 1.54616461 + -2.625

Combine like terms: 1.54616461 + -2.625 = -1.07883539
p = -1.07883539

Simplifying
p = -1.07883539


Subproblem 2
p + 2.625 = -1.54616461

Simplifying
p + 2.625 = -1.54616461

Reorder the terms:
2.625 + p = -1.54616461

Solving
2.625 + p = -1.54616461

Solving for variable 'p'.

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

Add '-2.625' to each side of the equation.
2.625 + -2.625 + p = -1.54616461 + -2.625

Combine like terms: 2.625 + -2.625 = 0.000
0.000 + p = -1.54616461 + -2.625
p = -1.54616461 + -2.625

Combine like terms: -1.54616461 + -2.625 = -4.17116461
p = -4.17116461

Simplifying
p = -4.17116461


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {-1.07883539, -4.17116461}
Solution
p = {-1.07883539, -4.17116461}

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