324x^2+162x+1=0

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Solution for 324x^2+162x+1=0 equation:


Simplifying
324x2 + 162x + 1 = 0

Reorder the terms:
1 + 162x + 324x2 = 0

Solving
1 + 162x + 324x2 = 0

Solving for variable 'x'.

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

Divide each side by '324'.
0.003086419753 + 0.5x + x2 = 0

Move the constant term to the right:

Add '-0.003086419753' to each side of the equation.
0.003086419753 + 0.5x + -0.003086419753 + x2 = 0 + -0.003086419753

Reorder the terms:
0.003086419753 + -0.003086419753 + 0.5x + x2 = 0 + -0.003086419753

Combine like terms: 0.003086419753 + -0.003086419753 = 0.000000000000
0.000000000000 + 0.5x + x2 = 0 + -0.003086419753
0.5x + x2 = 0 + -0.003086419753

Combine like terms: 0 + -0.003086419753 = -0.003086419753
0.5x + x2 = -0.003086419753

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 = -0.003086419753 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = -0.003086419753 + 0.0625

Combine like terms: -0.003086419753 + 0.0625 = 0.059413580247
0.0625 + 0.5x + x2 = 0.059413580247

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.059413580247

Calculate the square root of the right side: 0.243749011

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.243749011 and -0.243749011.

Subproblem 1

x + 0.25 = 0.243749011 Simplifying x + 0.25 = 0.243749011 Reorder the terms: 0.25 + x = 0.243749011 Solving 0.25 + x = 0.243749011 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 = 0.243749011 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.243749011 + -0.25 x = 0.243749011 + -0.25 Combine like terms: 0.243749011 + -0.25 = -0.006250989 x = -0.006250989 Simplifying x = -0.006250989

Subproblem 2

x + 0.25 = -0.243749011 Simplifying x + 0.25 = -0.243749011 Reorder the terms: 0.25 + x = -0.243749011 Solving 0.25 + x = -0.243749011 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 = -0.243749011 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.243749011 + -0.25 x = -0.243749011 + -0.25 Combine like terms: -0.243749011 + -0.25 = -0.493749011 x = -0.493749011 Simplifying x = -0.493749011

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.006250989, -0.493749011}

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