3x^2+13x+7=0

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Solution for 3x^2+13x+7=0 equation: 


Simplifying
3x2 + 13x + 7 = 0

Reorder the terms:
7 + 13x + 3x2 = 0

Solving
7 + 13x + 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'.
2.333333333 + 4.333333333x + x2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + 4.333333333x + -2.333333333 + x2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + 4.333333333x + x2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + 4.333333333x + x2 = 0 + -2.333333333
4.333333333x + x2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
4.333333333x + x2 = -2.333333333

The x term is 4.333333333x.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333x + 4.694444446 + x2 = -2.333333333 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333x + x2 = -2.333333333 + 4.694444446

Combine like terms: -2.333333333 + 4.694444446 = 2.361111113
4.694444446 + 4.333333333x + x2 = 2.361111113

Factor a perfect square on the left side:
(x + 2.166666667)(x + 2.166666667) = 2.361111113

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(x + 2.166666667) equal to 1.536590743 and -1.536590743.


Subproblem 1
x + 2.166666667 = 1.536590743

Simplifying
x + 2.166666667 = 1.536590743

Reorder the terms:
2.166666667 + x = 1.536590743

Solving
2.166666667 + x = 1.536590743

Solving for variable 'x'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = 1.536590743 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = 1.536590743 + -2.166666667
x = 1.536590743 + -2.166666667

Combine like terms: 1.536590743 + -2.166666667 = -0.630075924
x = -0.630075924

Simplifying
x = -0.630075924


Subproblem 2
x + 2.166666667 = -1.536590743

Simplifying
x + 2.166666667 = -1.536590743

Reorder the terms:
2.166666667 + x = -1.536590743

Solving
2.166666667 + x = -1.536590743

Solving for variable 'x'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = -1.536590743 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = -1.536590743 + -2.166666667
x = -1.536590743 + -2.166666667

Combine like terms: -1.536590743 + -2.166666667 = -3.70325741
x = -3.70325741

Simplifying
x = -3.70325741


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.630075924, -3.70325741}

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