3x^2-11x-5=0

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Solution for 3x^2-11x-5=0 equation: 


Simplifying
3x2 + -11x + -5 = 0

Reorder the terms:
-5 + -11x + 3x2 = 0

Solving
-5 + -11x + 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'.
-1.666666667 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -3.666666667x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -3.666666667x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -3.666666667x + x2 = 0 + 1.666666667
-3.666666667x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-3.666666667x + x2 = 1.666666667

The x term is -3.666666667x.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667x + 3.361111114 + x2 = 1.666666667 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = 1.666666667 + 3.361111114

Combine like terms: 1.666666667 + 3.361111114 = 5.027777781
3.361111114 + -3.666666667x + x2 = 5.027777781

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 5.027777781

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 2.242270675 and -2.242270675.


Subproblem 1
x + -1.833333334 = 2.242270675

Simplifying
x + -1.833333334 = 2.242270675

Reorder the terms:
-1.833333334 + x = 2.242270675

Solving
-1.833333334 + x = 2.242270675

Solving for variable 'x'.

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

Add '1.833333334' to each side of the equation.
-1.833333334 + 1.833333334 + x = 2.242270675 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + x = 2.242270675 + 1.833333334
x = 2.242270675 + 1.833333334

Combine like terms: 2.242270675 + 1.833333334 = 4.075604009
x = 4.075604009

Simplifying
x = 4.075604009


Subproblem 2
x + -1.833333334 = -2.242270675

Simplifying
x + -1.833333334 = -2.242270675

Reorder the terms:
-1.833333334 + x = -2.242270675

Solving
-1.833333334 + x = -2.242270675

Solving for variable 'x'.

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

Add '1.833333334' to each side of the equation.
-1.833333334 + 1.833333334 + x = -2.242270675 + 1.833333334

Combine like terms: -1.833333334 + 1.833333334 = 0.000000000
0.000000000 + x = -2.242270675 + 1.833333334
x = -2.242270675 + 1.833333334

Combine like terms: -2.242270675 + 1.833333334 = -0.408937341
x = -0.408937341

Simplifying
x = -0.408937341


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.075604009, -0.408937341}

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