7-4x=3x^2-6x+3

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Solution for 7-4x=3x^2-6x+3 equation: 


Simplifying
7 + -4x = 3x2 + -6x + 3

Reorder the terms:
7 + -4x = 3 + -6x + 3x2

Solving
7 + -4x = 3 + -6x + 3x2

Solving for variable 'x'.

Reorder the terms:
7 + -3 + -4x + 6x + -3x2 = 3 + -6x + 3x2 + -3 + 6x + -3x2

Combine like terms: 7 + -3 = 4
4 + -4x + 6x + -3x2 = 3 + -6x + 3x2 + -3 + 6x + -3x2

Combine like terms: -4x + 6x = 2x
4 + 2x + -3x2 = 3 + -6x + 3x2 + -3 + 6x + -3x2

Reorder the terms:
4 + 2x + -3x2 = 3 + -3 + -6x + 6x + 3x2 + -3x2

Combine like terms: 3 + -3 = 0
4 + 2x + -3x2 = 0 + -6x + 6x + 3x2 + -3x2
4 + 2x + -3x2 = -6x + 6x + 3x2 + -3x2

Combine like terms: -6x + 6x = 0
4 + 2x + -3x2 = 0 + 3x2 + -3x2
4 + 2x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
4 + 2x + -3x2 = 0

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

Divide each side by '-3'.
-1.333333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -0.6666666667x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -0.6666666667x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 1.333333333
-0.6666666667x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-0.6666666667x + x2 = 1.333333333

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 1.333333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 1.333333333 + 0.1111111112

Combine like terms: 1.333333333 + 0.1111111112 = 1.4444444442
0.1111111112 + -0.6666666667x + x2 = 1.4444444442

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.201850425 and -1.201850425.


Subproblem 1
x + -0.3333333334 = 1.201850425

Simplifying
x + -0.3333333334 = 1.201850425

Reorder the terms:
-0.3333333334 + x = 1.201850425

Solving
-0.3333333334 + x = 1.201850425

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 1.201850425 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 1.201850425 + 0.3333333334
x = 1.201850425 + 0.3333333334

Combine like terms: 1.201850425 + 0.3333333334 = 1.5351837584
x = 1.5351837584

Simplifying
x = 1.5351837584


Subproblem 2
x + -0.3333333334 = -1.201850425

Simplifying
x + -0.3333333334 = -1.201850425

Reorder the terms:
-0.3333333334 + x = -1.201850425

Solving
-0.3333333334 + x = -1.201850425

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -1.201850425 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -1.201850425 + 0.3333333334
x = -1.201850425 + 0.3333333334

Combine like terms: -1.201850425 + 0.3333333334 = -0.8685170916
x = -0.8685170916

Simplifying
x = -0.8685170916


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.5351837584, -0.8685170916}

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