X*(X+6)+X*(X+2)=2*X+14

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Solution for X*(X+6)+X*(X+2)=2*X+14 equation: 


Simplifying
X(X + 6) + X(X + 2) = 2X + 14

Reorder the terms:
X(6 + X) + X(X + 2) = 2X + 14
(6 * X + X * X) + X(X + 2) = 2X + 14
(6X + X2) + X(X + 2) = 2X + 14

Reorder the terms:
6X + X2 + X(2 + X) = 2X + 14
6X + X2 + (2 * X + X * X) = 2X + 14
6X + X2 + (2X + X2) = 2X + 14

Reorder the terms:
6X + 2X + X2 + X2 = 2X + 14

Combine like terms: 6X + 2X = 8X
8X + X2 + X2 = 2X + 14

Combine like terms: X2 + X2 = 2X2
8X + 2X2 = 2X + 14

Reorder the terms:
8X + 2X2 = 14 + 2X

Solving
8X + 2X2 = 14 + 2X

Solving for variable 'X'.

Reorder the terms:
-14 + 8X + -2X + 2X2 = 14 + 2X + -14 + -2X

Combine like terms: 8X + -2X = 6X
-14 + 6X + 2X2 = 14 + 2X + -14 + -2X

Reorder the terms:
-14 + 6X + 2X2 = 14 + -14 + 2X + -2X

Combine like terms: 14 + -14 = 0
-14 + 6X + 2X2 = 0 + 2X + -2X
-14 + 6X + 2X2 = 2X + -2X

Combine like terms: 2X + -2X = 0
-14 + 6X + 2X2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 3X + X2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + 3X + X2)' equal to zero and attempt to solve:

Simplifying
-7 + 3X + X2 = 0

Solving
-7 + 3X + X2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 3X + 7 + X2 = 0 + 7

Reorder the terms:
-7 + 7 + 3X + X2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 3X + X2 = 0 + 7
3X + X2 = 0 + 7

Combine like terms: 0 + 7 = 7
3X + X2 = 7

The X term is 3X.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3X + 2.25 + X2 = 7 + 2.25

Reorder the terms:
2.25 + 3X + X2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + 3X + X2 = 9.25

Factor a perfect square on the left side:
(X + 1.5)(X + 1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(X + 1.5) equal to 3.041381265 and -3.041381265.


Subproblem 1
X + 1.5 = 3.041381265

Simplifying
X + 1.5 = 3.041381265

Reorder the terms:
1.5 + X = 3.041381265

Solving
1.5 + X = 3.041381265

Solving for variable 'X'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + X = 3.041381265 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + X = 3.041381265 + -1.5
X = 3.041381265 + -1.5

Combine like terms: 3.041381265 + -1.5 = 1.541381265
X = 1.541381265

Simplifying
X = 1.541381265


Subproblem 2
X + 1.5 = -3.041381265

Simplifying
X + 1.5 = -3.041381265

Reorder the terms:
1.5 + X = -3.041381265

Solving
1.5 + X = -3.041381265

Solving for variable 'X'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + X = -3.041381265 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + X = -3.041381265 + -1.5
X = -3.041381265 + -1.5

Combine like terms: -3.041381265 + -1.5 = -4.541381265
X = -4.541381265

Simplifying
X = -4.541381265


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {1.541381265, -4.541381265}
Solution
X = {1.541381265, -4.541381265}

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