6x^2+17x+6=0

Solution for 6x^2+17x+6=0 equation:

Simplifying
6x2 + 17x + 6 = 0

Reorder the terms:
6 + 17x + 6x2 = 0

Solving
6 + 17x + 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'.
1 + 2.833333333x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.833333333x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.833333333x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.833333333x + x2 = 0 + -1
2.833333333x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.833333333x + x2 = -1

The x term is 2.833333333x.  Take half its coefficient (1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
2.833333333x + 2.006944445 + x2 = -1 + 2.006944445

Reorder the terms:
2.006944445 + 2.833333333x + x2 = -1 + 2.006944445

Combine like terms: -1 + 2.006944445 = 1.006944445
2.006944445 + 2.833333333x + x2 = 1.006944445

Factor a perfect square on the left side:
(x + 1.416666667)(x + 1.416666667) = 1.006944445

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + 1.416666667) equal to 1.003466215 and -1.003466215.

Subproblem 1
x + 1.416666667 = 1.003466215

Simplifying
x + 1.416666667 = 1.003466215

Reorder the terms:
1.416666667 + x = 1.003466215

Solving
1.416666667 + x = 1.003466215

Solving for variable 'x'.

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

Add '-1.416666667' to each side of the equation.
1.416666667 + -1.416666667 + x = 1.003466215 + -1.416666667

Combine like terms: 1.416666667 + -1.416666667 = 0.000000000
0.000000000 + x = 1.003466215 + -1.416666667
x = 1.003466215 + -1.416666667

Combine like terms: 1.003466215 + -1.416666667 = -0.413200452
x = -0.413200452

Simplifying
x = -0.413200452

Subproblem 2
x + 1.416666667 = -1.003466215

Simplifying
x + 1.416666667 = -1.003466215

Reorder the terms:
1.416666667 + x = -1.003466215

Solving
1.416666667 + x = -1.003466215

Solving for variable 'x'.

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

Add '-1.416666667' to each side of the equation.
1.416666667 + -1.416666667 + x = -1.003466215 + -1.416666667

Combine like terms: 1.416666667 + -1.416666667 = 0.000000000
0.000000000 + x = -1.003466215 + -1.416666667
x = -1.003466215 + -1.416666667

Combine like terms: -1.003466215 + -1.416666667 = -2.420132882
x = -2.420132882

Simplifying
x = -2.420132882

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.413200452, -2.420132882}