3x^2-19x+22=0

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Solution for 3x^2-19x+22=0 equation: 


Simplifying
3x2 + -19x + 22 = 0

Reorder the terms:
22 + -19x + 3x2 = 0

Solving
22 + -19x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
7.333333333 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '-7.333333333' to each side of the equation.
7.333333333 + -6.333333333x + -7.333333333 + x2 = 0 + -7.333333333

Reorder the terms:
7.333333333 + -7.333333333 + -6.333333333x + x2 = 0 + -7.333333333

Combine like terms: 7.333333333 + -7.333333333 = 0.000000000
0.000000000 + -6.333333333x + x2 = 0 + -7.333333333
-6.333333333x + x2 = 0 + -7.333333333

Combine like terms: 0 + -7.333333333 = -7.333333333
-6.333333333x + x2 = -7.333333333

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = -7.333333333 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = -7.333333333 + 10.02777778

Combine like terms: -7.333333333 + 10.02777778 = 2.694444447
10.02777778 + -6.333333333x + x2 = 2.694444447

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 2.694444447

Calculate the square root of the right side: 1.641476301

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 1.641476301 and -1.641476301.


Subproblem 1
x + -3.166666667 = 1.641476301

Simplifying
x + -3.166666667 = 1.641476301

Reorder the terms:
-3.166666667 + x = 1.641476301

Solving
-3.166666667 + x = 1.641476301

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = 1.641476301 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = 1.641476301 + 3.166666667
x = 1.641476301 + 3.166666667

Combine like terms: 1.641476301 + 3.166666667 = 4.808142968
x = 4.808142968

Simplifying
x = 4.808142968


Subproblem 2
x + -3.166666667 = -1.641476301

Simplifying
x + -3.166666667 = -1.641476301

Reorder the terms:
-3.166666667 + x = -1.641476301

Solving
-3.166666667 + x = -1.641476301

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = -1.641476301 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = -1.641476301 + 3.166666667
x = -1.641476301 + 3.166666667

Combine like terms: -1.641476301 + 3.166666667 = 1.525190366
x = 1.525190366

Simplifying
x = 1.525190366


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.808142968, 1.525190366}

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