(x-1)(3x+1)=5

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


Simplifying
(x + -1)(3x + 1) = 5

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

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

Multiply (-1 + x) * (1 + 3x)
(-1(1 + 3x) + x(1 + 3x)) = 5
((1 * -1 + 3x * -1) + x(1 + 3x)) = 5
((-1 + -3x) + x(1 + 3x)) = 5
(-1 + -3x + (1 * x + 3x * x)) = 5
(-1 + -3x + (1x + 3x2)) = 5

Combine like terms: -3x + 1x = -2x
(-1 + -2x + 3x2) = 5

Solving
-1 + -2x + 3x2 = 5

Solving for variable 'x'.

Reorder the terms:
-1 + -5 + -2x + 3x2 = 5 + -5

Combine like terms: -1 + -5 = -6
-6 + -2x + 3x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-6 + -2x + 3x2 = 0

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

Divide each side by '3'.
-2 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.6666666667x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.6666666667x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.6666666667x + x2 = 0 + 2
-0.6666666667x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.6666666667x + x2 = 2

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 2 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 2 + 0.1111111112

Combine like terms: 2 + 0.1111111112 = 2.1111111112
0.1111111112 + -0.6666666667x + x2 = 2.1111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 2.1111111112

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1

x + -0.3333333334 = 1.452966315 Simplifying x + -0.3333333334 = 1.452966315 Reorder the terms: -0.3333333334 + x = 1.452966315 Solving -0.3333333334 + x = 1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.452966315 + 0.3333333334 x = 1.452966315 + 0.3333333334 Combine like terms: 1.452966315 + 0.3333333334 = 1.7862996484 x = 1.7862996484 Simplifying x = 1.7862996484

Subproblem 2

x + -0.3333333334 = -1.452966315 Simplifying x + -0.3333333334 = -1.452966315 Reorder the terms: -0.3333333334 + x = -1.452966315 Solving -0.3333333334 + x = -1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.452966315 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.452966315 + 0.3333333334 x = -1.452966315 + 0.3333333334 Combine like terms: -1.452966315 + 0.3333333334 = -1.1196329816 x = -1.1196329816 Simplifying x = -1.1196329816

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7862996484, -1.1196329816}

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