3x^2-23x+13=0

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Solution for 3x^2-23x+13=0 equation: 


Simplifying
3x2 + -23x + 13 = 0

Reorder the terms:
13 + -23x + 3x2 = 0

Solving
13 + -23x + 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'.
4.333333333 + -7.666666667x + x2 = 0

Move the constant term to the right:

Add '-4.333333333' to each side of the equation.
4.333333333 + -7.666666667x + -4.333333333 + x2 = 0 + -4.333333333

Reorder the terms:
4.333333333 + -4.333333333 + -7.666666667x + x2 = 0 + -4.333333333

Combine like terms: 4.333333333 + -4.333333333 = 0.000000000
0.000000000 + -7.666666667x + x2 = 0 + -4.333333333
-7.666666667x + x2 = 0 + -4.333333333

Combine like terms: 0 + -4.333333333 = -4.333333333
-7.666666667x + x2 = -4.333333333

The x term is -7.666666667x.  Take half its coefficient (-3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
-7.666666667x + 14.69444445 + x2 = -4.333333333 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667x + x2 = -4.333333333 + 14.69444445

Combine like terms: -4.333333333 + 14.69444445 = 10.361111117
14.69444445 + -7.666666667x + x2 = 10.361111117

Factor a perfect square on the left side:
(x + -3.833333334)(x + -3.833333334) = 10.361111117

Calculate the square root of the right side: 3.218867987

Break this problem into two subproblems by setting 
(x + -3.833333334) equal to 3.218867987 and -3.218867987.


Subproblem 1
x + -3.833333334 = 3.218867987

Simplifying
x + -3.833333334 = 3.218867987

Reorder the terms:
-3.833333334 + x = 3.218867987

Solving
-3.833333334 + x = 3.218867987

Solving for variable 'x'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + x = 3.218867987 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = 3.218867987 + 3.833333334
x = 3.218867987 + 3.833333334

Combine like terms: 3.218867987 + 3.833333334 = 7.052201321
x = 7.052201321

Simplifying
x = 7.052201321


Subproblem 2
x + -3.833333334 = -3.218867987

Simplifying
x + -3.833333334 = -3.218867987

Reorder the terms:
-3.833333334 + x = -3.218867987

Solving
-3.833333334 + x = -3.218867987

Solving for variable 'x'.

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

Add '3.833333334' to each side of the equation.
-3.833333334 + 3.833333334 + x = -3.218867987 + 3.833333334

Combine like terms: -3.833333334 + 3.833333334 = 0.000000000
0.000000000 + x = -3.218867987 + 3.833333334
x = -3.218867987 + 3.833333334

Combine like terms: -3.218867987 + 3.833333334 = 0.614465347
x = 0.614465347

Simplifying
x = 0.614465347


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.052201321, 0.614465347}

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