x(6x+3)=10

Solution for x(6x+3)=10 equation:

Simplifying
x(6x + 3) = 10

Reorder the terms:
x(3 + 6x) = 10
(3 * x + 6x * x) = 10
(3x + 6x2) = 10

Solving
3x + 6x2 = 10

Solving for variable 'x'.

Reorder the terms:
-10 + 3x + 6x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 3x + 6x2 = 0

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

Divide each side by '6'.
-1.666666667 + 0.5x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 0.5x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 0.5x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 0.5x + x2 = 0 + 1.666666667
0.5x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
0.5x + x2 = 1.666666667

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 1.666666667 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 1.666666667 + 0.0625

Combine like terms: 1.666666667 + 0.0625 = 1.729166667
0.0625 + 0.5x + x2 = 1.729166667

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 1.729166667

Calculate the square root of the right side: 1.31497782

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.31497782 and -1.31497782.

Subproblem 1
x + 0.25 = 1.31497782

Simplifying
x + 0.25 = 1.31497782

Reorder the terms:
0.25 + x = 1.31497782

Solving
0.25 + x = 1.31497782

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 1.31497782 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 1.31497782 + -0.25
x = 1.31497782 + -0.25

Combine like terms: 1.31497782 + -0.25 = 1.06497782
x = 1.06497782

Simplifying
x = 1.06497782

Subproblem 2
x + 0.25 = -1.31497782

Simplifying
x + 0.25 = -1.31497782

Reorder the terms:
0.25 + x = -1.31497782

Solving
0.25 + x = -1.31497782

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -1.31497782 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -1.31497782 + -0.25
x = -1.31497782 + -0.25

Combine like terms: -1.31497782 + -0.25 = -1.56497782
x = -1.56497782

Simplifying
x = -1.56497782

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.06497782, -1.56497782}