12x^2-6x+42x-42=0

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Solution for 12x^2-6x+42x-42=0 equation: 


Simplifying
12x2 + -6x + 42x + -42 = 0

Reorder the terms:
-42 + -6x + 42x + 12x2 = 0

Combine like terms: -6x + 42x = 36x
-42 + 36x + 12x2 = 0

Solving
-42 + 36x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-7 + 6x + 2x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-7 + 6x + 2x2)' equal to zero and attempt to solve:

Simplifying
-7 + 6x + 2x2 = 0

Solving
-7 + 6x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + 3x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 3x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 3x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 3x + x2 = 0 + 3.5
3x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
3x + x2 = 3.5

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 3.5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 3.5 + 2.25

Combine like terms: 3.5 + 2.25 = 5.75
2.25 + 3x + x2 = 5.75

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.397915762 and -2.397915762.


Subproblem 1
x + 1.5 = 2.397915762

Simplifying
x + 1.5 = 2.397915762

Reorder the terms:
1.5 + x = 2.397915762

Solving
1.5 + x = 2.397915762

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.397915762 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.397915762 + -1.5
x = 2.397915762 + -1.5

Combine like terms: 2.397915762 + -1.5 = 0.897915762
x = 0.897915762

Simplifying
x = 0.897915762


Subproblem 2
x + 1.5 = -2.397915762

Simplifying
x + 1.5 = -2.397915762

Reorder the terms:
1.5 + x = -2.397915762

Solving
1.5 + x = -2.397915762

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.397915762 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.397915762 + -1.5
x = -2.397915762 + -1.5

Combine like terms: -2.397915762 + -1.5 = -3.897915762
x = -3.897915762

Simplifying
x = -3.897915762


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.897915762, -3.897915762}
Solution
x = {0.897915762, -3.897915762}

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