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

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


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

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

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

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

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

Solving
2 + 7x + 3x2 = 1

Solving for variable 'x'.

Reorder the terms:
2 + -1 + 7x + 3x2 = 1 + -1

Combine like terms: 2 + -1 = 1
1 + 7x + 3x2 = 1 + -1

Combine like terms: 1 + -1 = 0
1 + 7x + 3x2 = 0

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

Divide each side by '3'.
0.3333333333 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 2.333333333x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 2.333333333x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 2.333333333x + x2 = 0 + -0.3333333333
2.333333333x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
2.333333333x + x2 = -0.3333333333

The x term is 2.333333333x.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333x + 1.361111112 + x2 = -0.3333333333 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = -0.3333333333 + 1.361111112

Combine like terms: -0.3333333333 + 1.361111112 = 1.0277777787
1.361111112 + 2.333333333x + x2 = 1.0277777787

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 1.0277777787

Calculate the square root of the right side: 1.013793756

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 1.013793756 and -1.013793756.

Subproblem 1

x + 1.166666667 = 1.013793756 Simplifying x + 1.166666667 = 1.013793756 Reorder the terms: 1.166666667 + x = 1.013793756 Solving 1.166666667 + x = 1.013793756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 1.013793756 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 1.013793756 + -1.166666667 x = 1.013793756 + -1.166666667 Combine like terms: 1.013793756 + -1.166666667 = -0.152872911 x = -0.152872911 Simplifying x = -0.152872911

Subproblem 2

x + 1.166666667 = -1.013793756 Simplifying x + 1.166666667 = -1.013793756 Reorder the terms: 1.166666667 + x = -1.013793756 Solving 1.166666667 + x = -1.013793756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -1.013793756 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -1.013793756 + -1.166666667 x = -1.013793756 + -1.166666667 Combine like terms: -1.013793756 + -1.166666667 = -2.180460423 x = -2.180460423 Simplifying x = -2.180460423

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.152872911, -2.180460423}

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