(5X-1)(3X+1)=67

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Solution for (5X-1)(3X+1)=67 equation: 


Simplifying
(5X + -1)(3X + 1) = 67

Reorder the terms:
(-1 + 5X)(3X + 1) = 67

Reorder the terms:
(-1 + 5X)(1 + 3X) = 67

Multiply (-1 + 5X) * (1 + 3X)
(-1(1 + 3X) + 5X * (1 + 3X)) = 67
((1 * -1 + 3X * -1) + 5X * (1 + 3X)) = 67
((-1 + -3X) + 5X * (1 + 3X)) = 67
(-1 + -3X + (1 * 5X + 3X * 5X)) = 67
(-1 + -3X + (5X + 15X2)) = 67

Combine like terms: -3X + 5X = 2X
(-1 + 2X + 15X2) = 67

Solving
-1 + 2X + 15X2 = 67

Solving for variable 'X'.

Reorder the terms:
-1 + -67 + 2X + 15X2 = 67 + -67

Combine like terms: -1 + -67 = -68
-68 + 2X + 15X2 = 67 + -67

Combine like terms: 67 + -67 = 0
-68 + 2X + 15X2 = 0

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

Divide each side by '15'.
-4.533333333 + 0.1333333333X + X2 = 0

Move the constant term to the right:

Add '4.533333333' to each side of the equation.
-4.533333333 + 0.1333333333X + 4.533333333 + X2 = 0 + 4.533333333

Reorder the terms:
-4.533333333 + 4.533333333 + 0.1333333333X + X2 = 0 + 4.533333333

Combine like terms: -4.533333333 + 4.533333333 = 0.000000000
0.000000000 + 0.1333333333X + X2 = 0 + 4.533333333
0.1333333333X + X2 = 0 + 4.533333333

Combine like terms: 0 + 4.533333333 = 4.533333333
0.1333333333X + X2 = 4.533333333

The X term is 0.1333333333X.  Take half its coefficient (0.06666666665).
Square it (0.004444444442) and add it to both sides.

Add '0.004444444442' to each side of the equation.
0.1333333333X + 0.004444444442 + X2 = 4.533333333 + 0.004444444442

Reorder the terms:
0.004444444442 + 0.1333333333X + X2 = 4.533333333 + 0.004444444442

Combine like terms: 4.533333333 + 0.004444444442 = 4.537777777442
0.004444444442 + 0.1333333333X + X2 = 4.537777777442

Factor a perfect square on the left side:
(X + 0.06666666665)(X + 0.06666666665) = 4.537777777442

Calculate the square root of the right side: 2.130206041

Break this problem into two subproblems by setting 
(X + 0.06666666665) equal to 2.130206041 and -2.130206041.


Subproblem 1
X + 0.06666666665 = 2.130206041

Simplifying
X + 0.06666666665 = 2.130206041

Reorder the terms:
0.06666666665 + X = 2.130206041

Solving
0.06666666665 + X = 2.130206041

Solving for variable 'X'.

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

Add '-0.06666666665' to each side of the equation.
0.06666666665 + -0.06666666665 + X = 2.130206041 + -0.06666666665

Combine like terms: 0.06666666665 + -0.06666666665 = 0.00000000000
0.00000000000 + X = 2.130206041 + -0.06666666665
X = 2.130206041 + -0.06666666665

Combine like terms: 2.130206041 + -0.06666666665 = 2.06353937435
X = 2.06353937435

Simplifying
X = 2.06353937435


Subproblem 2
X + 0.06666666665 = -2.130206041

Simplifying
X + 0.06666666665 = -2.130206041

Reorder the terms:
0.06666666665 + X = -2.130206041

Solving
0.06666666665 + X = -2.130206041

Solving for variable 'X'.

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

Add '-0.06666666665' to each side of the equation.
0.06666666665 + -0.06666666665 + X = -2.130206041 + -0.06666666665

Combine like terms: 0.06666666665 + -0.06666666665 = 0.00000000000
0.00000000000 + X = -2.130206041 + -0.06666666665
X = -2.130206041 + -0.06666666665

Combine like terms: -2.130206041 + -0.06666666665 = -2.19687270765
X = -2.19687270765

Simplifying
X = -2.19687270765


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {2.06353937435, -2.19687270765}

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