(4X-1)(3X+2)=(3-X)(2X+5)

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Solution for (4X-1)(3X+2)=(3-X)(2X+5) equation: 


Simplifying
(4X + -1)(3X + 2) = (3 + -1X)(2X + 5)

Reorder the terms:
(-1 + 4X)(3X + 2) = (3 + -1X)(2X + 5)

Reorder the terms:
(-1 + 4X)(2 + 3X) = (3 + -1X)(2X + 5)

Multiply (-1 + 4X) * (2 + 3X)
(-1(2 + 3X) + 4X * (2 + 3X)) = (3 + -1X)(2X + 5)
((2 * -1 + 3X * -1) + 4X * (2 + 3X)) = (3 + -1X)(2X + 5)
((-2 + -3X) + 4X * (2 + 3X)) = (3 + -1X)(2X + 5)
(-2 + -3X + (2 * 4X + 3X * 4X)) = (3 + -1X)(2X + 5)
(-2 + -3X + (8X + 12X2)) = (3 + -1X)(2X + 5)

Combine like terms: -3X + 8X = 5X
(-2 + 5X + 12X2) = (3 + -1X)(2X + 5)

Reorder the terms:
-2 + 5X + 12X2 = (3 + -1X)(5 + 2X)

Multiply (3 + -1X) * (5 + 2X)
-2 + 5X + 12X2 = (3(5 + 2X) + -1X * (5 + 2X))
-2 + 5X + 12X2 = ((5 * 3 + 2X * 3) + -1X * (5 + 2X))
-2 + 5X + 12X2 = ((15 + 6X) + -1X * (5 + 2X))
-2 + 5X + 12X2 = (15 + 6X + (5 * -1X + 2X * -1X))
-2 + 5X + 12X2 = (15 + 6X + (-5X + -2X2))

Combine like terms: 6X + -5X = 1X
-2 + 5X + 12X2 = (15 + 1X + -2X2)

Solving
-2 + 5X + 12X2 = 15 + 1X + -2X2

Solving for variable 'X'.

Reorder the terms:
-2 + -15 + 5X + -1X + 12X2 + 2X2 = 15 + 1X + -2X2 + -15 + -1X + 2X2

Combine like terms: -2 + -15 = -17
-17 + 5X + -1X + 12X2 + 2X2 = 15 + 1X + -2X2 + -15 + -1X + 2X2

Combine like terms: 5X + -1X = 4X
-17 + 4X + 12X2 + 2X2 = 15 + 1X + -2X2 + -15 + -1X + 2X2

Combine like terms: 12X2 + 2X2 = 14X2
-17 + 4X + 14X2 = 15 + 1X + -2X2 + -15 + -1X + 2X2

Reorder the terms:
-17 + 4X + 14X2 = 15 + -15 + 1X + -1X + -2X2 + 2X2

Combine like terms: 15 + -15 = 0
-17 + 4X + 14X2 = 0 + 1X + -1X + -2X2 + 2X2
-17 + 4X + 14X2 = 1X + -1X + -2X2 + 2X2

Combine like terms: 1X + -1X = 0
-17 + 4X + 14X2 = 0 + -2X2 + 2X2
-17 + 4X + 14X2 = -2X2 + 2X2

Combine like terms: -2X2 + 2X2 = 0
-17 + 4X + 14X2 = 0

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

Divide each side by '14'.
-1.214285714 + 0.2857142857X + X2 = 0

Move the constant term to the right:

Add '1.214285714' to each side of the equation.
-1.214285714 + 0.2857142857X + 1.214285714 + X2 = 0 + 1.214285714

Reorder the terms:
-1.214285714 + 1.214285714 + 0.2857142857X + X2 = 0 + 1.214285714

Combine like terms: -1.214285714 + 1.214285714 = 0.000000000
0.000000000 + 0.2857142857X + X2 = 0 + 1.214285714
0.2857142857X + X2 = 0 + 1.214285714

Combine like terms: 0 + 1.214285714 = 1.214285714
0.2857142857X + X2 = 1.214285714

The X term is 0.2857142857X.  Take half its coefficient (0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
0.2857142857X + 0.02040816328 + X2 = 1.214285714 + 0.02040816328

Reorder the terms:
0.02040816328 + 0.2857142857X + X2 = 1.214285714 + 0.02040816328

Combine like terms: 1.214285714 + 0.02040816328 = 1.23469387728
0.02040816328 + 0.2857142857X + X2 = 1.23469387728

Factor a perfect square on the left side:
(X + 0.1428571429)(X + 0.1428571429) = 1.23469387728

Calculate the square root of the right side: 1.111167799

Break this problem into two subproblems by setting 
(X + 0.1428571429) equal to 1.111167799 and -1.111167799.


Subproblem 1
X + 0.1428571429 = 1.111167799

Simplifying
X + 0.1428571429 = 1.111167799

Reorder the terms:
0.1428571429 + X = 1.111167799

Solving
0.1428571429 + X = 1.111167799

Solving for variable 'X'.

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

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -0.1428571429 + X = 1.111167799 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + X = 1.111167799 + -0.1428571429
X = 1.111167799 + -0.1428571429

Combine like terms: 1.111167799 + -0.1428571429 = 0.9683106561
X = 0.9683106561

Simplifying
X = 0.9683106561


Subproblem 2
X + 0.1428571429 = -1.111167799

Simplifying
X + 0.1428571429 = -1.111167799

Reorder the terms:
0.1428571429 + X = -1.111167799

Solving
0.1428571429 + X = -1.111167799

Solving for variable 'X'.

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

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -0.1428571429 + X = -1.111167799 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + X = -1.111167799 + -0.1428571429
X = -1.111167799 + -0.1428571429

Combine like terms: -1.111167799 + -0.1428571429 = -1.2540249419
X = -1.2540249419

Simplifying
X = -1.2540249419


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {0.9683106561, -1.2540249419}

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