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

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


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

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

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

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

Combine like terms: 1x + -9x = -8x
(-3 + -8x + 3x2) = 1

Solving
-3 + -8x + 3x2 = 1

Solving for variable 'x'.

Reorder the terms:
-3 + -1 + -8x + 3x2 = 1 + -1

Combine like terms: -3 + -1 = -4
-4 + -8x + 3x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-4 + -8x + 3x2 = 0

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

Divide each side by '3'.
-1.333333333 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -2.666666667x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -2.666666667x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -2.666666667x + x2 = 0 + 1.333333333
-2.666666667x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-2.666666667x + x2 = 1.333333333

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 1.333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 1.333333333 + 1.777777780

Combine like terms: 1.333333333 + 1.777777780 = 3.111111113
1.777777780 + -2.666666667x + x2 = 3.111111113

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 3.111111113

Calculate the square root of the right side: 1.763834208

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.763834208 and -1.763834208.

Subproblem 1

x + -1.333333334 = 1.763834208 Simplifying x + -1.333333334 = 1.763834208 Reorder the terms: -1.333333334 + x = 1.763834208 Solving -1.333333334 + x = 1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.763834208 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.763834208 + 1.333333334 x = 1.763834208 + 1.333333334 Combine like terms: 1.763834208 + 1.333333334 = 3.097167542 x = 3.097167542 Simplifying x = 3.097167542

Subproblem 2

x + -1.333333334 = -1.763834208 Simplifying x + -1.333333334 = -1.763834208 Reorder the terms: -1.333333334 + x = -1.763834208 Solving -1.333333334 + x = -1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.763834208 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.763834208 + 1.333333334 x = -1.763834208 + 1.333333334 Combine like terms: -1.763834208 + 1.333333334 = -0.430500874 x = -0.430500874 Simplifying x = -0.430500874

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.097167542, -0.430500874}

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