6x^2+18x+1=0

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Solution for 6x^2+18x+1=0 equation:


Simplifying
6x2 + 18x + 1 = 0

Reorder the terms:
1 + 18x + 6x2 = 0

Solving
1 + 18x + 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'.
0.1666666667 + 3x + x2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + 3x + -0.1666666667 + x2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + 3x + x2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + 3x + x2 = 0 + -0.1666666667
3x + x2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
3x + x2 = -0.1666666667

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = -0.1666666667 + 2.25

Reorder the terms:
2.25 + 3x + x2 = -0.1666666667 + 2.25

Combine like terms: -0.1666666667 + 2.25 = 2.0833333333
2.25 + 3x + x2 = 2.0833333333

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 2.0833333333

Calculate the square root of the right side: 1.443375673

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.443375673 and -1.443375673.

Subproblem 1

x + 1.5 = 1.443375673 Simplifying x + 1.5 = 1.443375673 Reorder the terms: 1.5 + x = 1.443375673 Solving 1.5 + x = 1.443375673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.443375673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.443375673 + -1.5 x = 1.443375673 + -1.5 Combine like terms: 1.443375673 + -1.5 = -0.056624327 x = -0.056624327 Simplifying x = -0.056624327

Subproblem 2

x + 1.5 = -1.443375673 Simplifying x + 1.5 = -1.443375673 Reorder the terms: 1.5 + x = -1.443375673 Solving 1.5 + x = -1.443375673 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.443375673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.443375673 + -1.5 x = -1.443375673 + -1.5 Combine like terms: -1.443375673 + -1.5 = -2.943375673 x = -2.943375673 Simplifying x = -2.943375673

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.056624327, -2.943375673}

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