3x(2x-3)-x=31

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Solution for 3x(2x-3)-x=31 equation:


Simplifying
3x(2x + -3) + -1x = 31

Reorder the terms:
3x(-3 + 2x) + -1x = 31
(-3 * 3x + 2x * 3x) + -1x = 31
(-9x + 6x2) + -1x = 31

Reorder the terms:
-9x + -1x + 6x2 = 31

Combine like terms: -9x + -1x = -10x
-10x + 6x2 = 31

Solving
-10x + 6x2 = 31

Solving for variable 'x'.

Reorder the terms:
-31 + -10x + 6x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + -10x + 6x2 = 0

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

Divide each side by '6'.
-5.166666667 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '5.166666667' to each side of the equation.
-5.166666667 + -1.666666667x + 5.166666667 + x2 = 0 + 5.166666667

Reorder the terms:
-5.166666667 + 5.166666667 + -1.666666667x + x2 = 0 + 5.166666667

Combine like terms: -5.166666667 + 5.166666667 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 5.166666667
-1.666666667x + x2 = 0 + 5.166666667

Combine like terms: 0 + 5.166666667 = 5.166666667
-1.666666667x + x2 = 5.166666667

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 = 5.166666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 5.166666667 + 0.6944444447

Combine like terms: 5.166666667 + 0.6944444447 = 5.8611111117
0.6944444447 + -1.666666667x + x2 = 5.8611111117

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 5.8611111117

Calculate the square root of the right side: 2.420973175

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.420973175 and -2.420973175.

Subproblem 1

x + -0.8333333335 = 2.420973175 Simplifying x + -0.8333333335 = 2.420973175 Reorder the terms: -0.8333333335 + x = 2.420973175 Solving -0.8333333335 + x = 2.420973175 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 = 2.420973175 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.420973175 + 0.8333333335 x = 2.420973175 + 0.8333333335 Combine like terms: 2.420973175 + 0.8333333335 = 3.2543065085 x = 3.2543065085 Simplifying x = 3.2543065085

Subproblem 2

x + -0.8333333335 = -2.420973175 Simplifying x + -0.8333333335 = -2.420973175 Reorder the terms: -0.8333333335 + x = -2.420973175 Solving -0.8333333335 + x = -2.420973175 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 = -2.420973175 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.420973175 + 0.8333333335 x = -2.420973175 + 0.8333333335 Combine like terms: -2.420973175 + 0.8333333335 = -1.5876398415 x = -1.5876398415 Simplifying x = -1.5876398415

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.2543065085, -1.5876398415}

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