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 = 1.694444447
3.361111114 + 3.666666667x + x2 = 1.694444447

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 1.694444447

Calculate the square root of the right side: 1.30170828

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 1.30170828 and -1.30170828.


Subproblem 1
x + 1.833333334 = 1.30170828

Simplifying
x + 1.833333334 = 1.30170828

Reorder the terms:
1.833333334 + x = 1.30170828

Solving
1.833333334 + x = 1.30170828

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 = 1.30170828 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + x = 1.30170828 + -1.833333334
x = 1.30170828 + -1.833333334

Combine like terms: 1.30170828 + -1.833333334 = -0.531625054
x = -0.531625054

Simplifying
x = -0.531625054


Subproblem 2
x + 1.833333334 = -1.30170828

Simplifying
x + 1.833333334 = -1.30170828

Reorder the terms:
1.833333334 + x = -1.30170828

Solving
1.833333334 + x = -1.30170828

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 = -1.30170828 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + x = -1.30170828 + -1.833333334
x = -1.30170828 + -1.833333334

Combine like terms: -1.30170828 + -1.833333334 = -3.135041614
x = -3.135041614

Simplifying
x = -3.135041614


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.531625054, -3.135041614}

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