6x^2+21x+13=0

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Solution for 6x^2+21x+13=0 equation: 


Simplifying
6x2 + 21x + 13 = 0

Reorder the terms:
13 + 21x + 6x2 = 0

Solving
13 + 21x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
2.166666667 + 3.5x + x2 = 0

Move the constant term to the right:

Add '-2.166666667' to each side of the equation.
2.166666667 + 3.5x + -2.166666667 + x2 = 0 + -2.166666667

Reorder the terms:
2.166666667 + -2.166666667 + 3.5x + x2 = 0 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + 3.5x + x2 = 0 + -2.166666667
3.5x + x2 = 0 + -2.166666667

Combine like terms: 0 + -2.166666667 = -2.166666667
3.5x + x2 = -2.166666667

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = -2.166666667 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = -2.166666667 + 3.0625

Combine like terms: -2.166666667 + 3.0625 = 0.895833333
3.0625 + 3.5x + x2 = 0.895833333

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 0.895833333

Calculate the square root of the right side: 0.946484724

Break this problem into two subproblems by setting 
(x + 1.75) equal to 0.946484724 and -0.946484724.


Subproblem 1
x + 1.75 = 0.946484724

Simplifying
x + 1.75 = 0.946484724

Reorder the terms:
1.75 + x = 0.946484724

Solving
1.75 + x = 0.946484724

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 0.946484724 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 0.946484724 + -1.75
x = 0.946484724 + -1.75

Combine like terms: 0.946484724 + -1.75 = -0.803515276
x = -0.803515276

Simplifying
x = -0.803515276


Subproblem 2
x + 1.75 = -0.946484724

Simplifying
x + 1.75 = -0.946484724

Reorder the terms:
1.75 + x = -0.946484724

Solving
1.75 + x = -0.946484724

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -0.946484724 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -0.946484724 + -1.75
x = -0.946484724 + -1.75

Combine like terms: -0.946484724 + -1.75 = -2.696484724
x = -2.696484724

Simplifying
x = -2.696484724


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.803515276, -2.696484724}

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