(3x+1)(x-1)-3(x+2)=9

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Solution for (3x+1)(x-1)-3(x+2)=9 equation:


Simplifying
(3x + 1)(x + -1) + -3(x + 2) = 9

Reorder the terms:
(1 + 3x)(x + -1) + -3(x + 2) = 9

Reorder the terms:
(1 + 3x)(-1 + x) + -3(x + 2) = 9

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

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

Reorder the terms:
-1 + -2x + 3x2 + -3(2 + x) = 9
-1 + -2x + 3x2 + (2 * -3 + x * -3) = 9
-1 + -2x + 3x2 + (-6 + -3x) = 9

Reorder the terms:
-1 + -6 + -2x + -3x + 3x2 = 9

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

Combine like terms: -2x + -3x = -5x
-7 + -5x + 3x2 = 9

Solving
-7 + -5x + 3x2 = 9

Solving for variable 'x'.

Reorder the terms:
-7 + -9 + -5x + 3x2 = 9 + -9

Combine like terms: -7 + -9 = -16
-16 + -5x + 3x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-16 + -5x + 3x2 = 0

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

Divide each side by '3'.
-5.333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + -1.666666667x + 5.333333333 + x2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + -1.666666667x + x2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 5.333333333
-1.666666667x + x2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
-1.666666667x + x2 = 5.333333333

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 5.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 5.333333333 + 0.6944444447

Combine like terms: 5.333333333 + 0.6944444447 = 6.0277777777
0.6944444447 + -1.666666667x + x2 = 6.0277777777

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 6.0277777777

Calculate the square root of the right side: 2.45515331

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.45515331 and -2.45515331.

Subproblem 1

x + -0.8333333335 = 2.45515331 Simplifying x + -0.8333333335 = 2.45515331 Reorder the terms: -0.8333333335 + x = 2.45515331 Solving -0.8333333335 + x = 2.45515331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.45515331 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.45515331 + 0.8333333335 x = 2.45515331 + 0.8333333335 Combine like terms: 2.45515331 + 0.8333333335 = 3.2884866435 x = 3.2884866435 Simplifying x = 3.2884866435

Subproblem 2

x + -0.8333333335 = -2.45515331 Simplifying x + -0.8333333335 = -2.45515331 Reorder the terms: -0.8333333335 + x = -2.45515331 Solving -0.8333333335 + x = -2.45515331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.45515331 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.45515331 + 0.8333333335 x = -2.45515331 + 0.8333333335 Combine like terms: -2.45515331 + 0.8333333335 = -1.6218199765 x = -1.6218199765 Simplifying x = -1.6218199765

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.2884866435, -1.6218199765}

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