22(x)=3x^2-4x+2

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Solution for 22(x)=3x^2-4x+2 equation: 


Simplifying
22(x) = 3x2 + -4x + 2

Reorder the terms:
22x = 2 + -4x + 3x2

Solving
22x = 2 + -4x + 3x2

Solving for variable 'x'.

Reorder the terms:
-2 + 22x + 4x + -3x2 = 2 + -4x + 3x2 + -2 + 4x + -3x2

Combine like terms: 22x + 4x = 26x
-2 + 26x + -3x2 = 2 + -4x + 3x2 + -2 + 4x + -3x2

Reorder the terms:
-2 + 26x + -3x2 = 2 + -2 + -4x + 4x + 3x2 + -3x2

Combine like terms: 2 + -2 = 0
-2 + 26x + -3x2 = 0 + -4x + 4x + 3x2 + -3x2
-2 + 26x + -3x2 = -4x + 4x + 3x2 + -3x2

Combine like terms: -4x + 4x = 0
-2 + 26x + -3x2 = 0 + 3x2 + -3x2
-2 + 26x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-2 + 26x + -3x2 = 0

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

Divide each side by '-3'.
0.6666666667 + -8.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -8.666666667x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -8.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -8.666666667x + x2 = 0 + -0.6666666667
-8.666666667x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-8.666666667x + x2 = -0.6666666667

The x term is -8.666666667x.  Take half its coefficient (-4.333333334).
Square it (18.77777778) and add it to both sides.

Add '18.77777778' to each side of the equation.
-8.666666667x + 18.77777778 + x2 = -0.6666666667 + 18.77777778

Reorder the terms:
18.77777778 + -8.666666667x + x2 = -0.6666666667 + 18.77777778

Combine like terms: -0.6666666667 + 18.77777778 = 18.1111111133
18.77777778 + -8.666666667x + x2 = 18.1111111133

Factor a perfect square on the left side:
(x + -4.333333334)(x + -4.333333334) = 18.1111111133

Calculate the square root of the right side: 4.255715112

Break this problem into two subproblems by setting 
(x + -4.333333334) equal to 4.255715112 and -4.255715112.


Subproblem 1
x + -4.333333334 = 4.255715112

Simplifying
x + -4.333333334 = 4.255715112

Reorder the terms:
-4.333333334 + x = 4.255715112

Solving
-4.333333334 + x = 4.255715112

Solving for variable 'x'.

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

Add '4.333333334' to each side of the equation.
-4.333333334 + 4.333333334 + x = 4.255715112 + 4.333333334

Combine like terms: -4.333333334 + 4.333333334 = 0.000000000
0.000000000 + x = 4.255715112 + 4.333333334
x = 4.255715112 + 4.333333334

Combine like terms: 4.255715112 + 4.333333334 = 8.589048446
x = 8.589048446

Simplifying
x = 8.589048446


Subproblem 2
x + -4.333333334 = -4.255715112

Simplifying
x + -4.333333334 = -4.255715112

Reorder the terms:
-4.333333334 + x = -4.255715112

Solving
-4.333333334 + x = -4.255715112

Solving for variable 'x'.

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

Add '4.333333334' to each side of the equation.
-4.333333334 + 4.333333334 + x = -4.255715112 + 4.333333334

Combine like terms: -4.333333334 + 4.333333334 = 0.000000000
0.000000000 + x = -4.255715112 + 4.333333334
x = -4.255715112 + 4.333333334

Combine like terms: -4.255715112 + 4.333333334 = 0.077618222
x = 0.077618222

Simplifying
x = 0.077618222


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.589048446, 0.077618222}

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