(3x+1)(2x-3)=1

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Solution for (3x+1)(2x-3)=1 equation:


Simplifying
(3x + 1)(2x + -3) = 1

Reorder the terms:
(1 + 3x)(2x + -3) = 1

Reorder the terms:
(1 + 3x)(-3 + 2x) = 1

Multiply (1 + 3x) * (-3 + 2x)
(1(-3 + 2x) + 3x * (-3 + 2x)) = 1
((-3 * 1 + 2x * 1) + 3x * (-3 + 2x)) = 1
((-3 + 2x) + 3x * (-3 + 2x)) = 1
(-3 + 2x + (-3 * 3x + 2x * 3x)) = 1
(-3 + 2x + (-9x + 6x2)) = 1

Combine like terms: 2x + -9x = -7x
(-3 + -7x + 6x2) = 1

Solving
-3 + -7x + 6x2 = 1

Solving for variable 'x'.

Reorder the terms:
-3 + -1 + -7x + 6x2 = 1 + -1

Combine like terms: -3 + -1 = -4
-4 + -7x + 6x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-4 + -7x + 6x2 = 0

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

Divide each side by '6'.
-0.6666666667 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -1.166666667x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -1.166666667x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -1.166666667x + x2 = 0 + 0.6666666667
-1.166666667x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-1.166666667x + x2 = 0.6666666667

The x term is -1.166666667x.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667x + 0.3402777780 + x2 = 0.6666666667 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 0.6666666667 + 0.3402777780

Combine like terms: 0.6666666667 + 0.3402777780 = 1.0069444447
0.3402777780 + -1.166666667x + x2 = 1.0069444447

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 1.0069444447

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 1.003466215 and -1.003466215.

Subproblem 1

x + -0.5833333335 = 1.003466215 Simplifying x + -0.5833333335 = 1.003466215 Reorder the terms: -0.5833333335 + x = 1.003466215 Solving -0.5833333335 + x = 1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = 1.003466215 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.003466215 + 0.5833333335 x = 1.003466215 + 0.5833333335 Combine like terms: 1.003466215 + 0.5833333335 = 1.5867995485 x = 1.5867995485 Simplifying x = 1.5867995485

Subproblem 2

x + -0.5833333335 = -1.003466215 Simplifying x + -0.5833333335 = -1.003466215 Reorder the terms: -0.5833333335 + x = -1.003466215 Solving -0.5833333335 + x = -1.003466215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = -1.003466215 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.003466215 + 0.5833333335 x = -1.003466215 + 0.5833333335 Combine like terms: -1.003466215 + 0.5833333335 = -0.4201328815 x = -0.4201328815 Simplifying x = -0.4201328815

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.5867995485, -0.4201328815}

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