3x(2x-2)=22-x

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Solution for 3x(2x-2)=22-x equation:


Simplifying
3x(2x + -2) = 22 + -1x

Reorder the terms:
3x(-2 + 2x) = 22 + -1x
(-2 * 3x + 2x * 3x) = 22 + -1x
(-6x + 6x2) = 22 + -1x

Solving
-6x + 6x2 = 22 + -1x

Solving for variable 'x'.

Reorder the terms:
-22 + -6x + x + 6x2 = 22 + -1x + -22 + x

Combine like terms: -6x + x = -5x
-22 + -5x + 6x2 = 22 + -1x + -22 + x

Reorder the terms:
-22 + -5x + 6x2 = 22 + -22 + -1x + x

Combine like terms: 22 + -22 = 0
-22 + -5x + 6x2 = 0 + -1x + x
-22 + -5x + 6x2 = -1x + x

Combine like terms: -1x + x = 0
-22 + -5x + 6x2 = 0

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

Divide each side by '6'.
-3.666666667 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -0.8333333333x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -0.8333333333x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -0.8333333333x + x2 = 0 + 3.666666667
-0.8333333333x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-0.8333333333x + x2 = 3.666666667

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 3.666666667 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 3.666666667 + 0.1736111111

Combine like terms: 3.666666667 + 0.1736111111 = 3.8402777781
0.1736111111 + -0.8333333333x + x2 = 3.8402777781

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 3.8402777781

Calculate the square root of the right side: 1.959662669

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 1.959662669 and -1.959662669.

Subproblem 1

x + -0.4166666667 = 1.959662669 Simplifying x + -0.4166666667 = 1.959662669 Reorder the terms: -0.4166666667 + x = 1.959662669 Solving -0.4166666667 + x = 1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 1.959662669 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 1.959662669 + 0.4166666667 x = 1.959662669 + 0.4166666667 Combine like terms: 1.959662669 + 0.4166666667 = 2.3763293357 x = 2.3763293357 Simplifying x = 2.3763293357

Subproblem 2

x + -0.4166666667 = -1.959662669 Simplifying x + -0.4166666667 = -1.959662669 Reorder the terms: -0.4166666667 + x = -1.959662669 Solving -0.4166666667 + x = -1.959662669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -1.959662669 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -1.959662669 + 0.4166666667 x = -1.959662669 + 0.4166666667 Combine like terms: -1.959662669 + 0.4166666667 = -1.5429960023 x = -1.5429960023 Simplifying x = -1.5429960023

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.3763293357, -1.5429960023}

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