3X^2+20X+1=0

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Solution for 3X^2+20X+1=0 equation: 


Simplifying
3X2 + 20X + 1 = 0

Reorder the terms:
1 + 20X + 3X2 = 0

Solving
1 + 20X + 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'.
0.3333333333 + 6.666666667X + X2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 6.666666667X + -0.3333333333 + X2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 6.666666667X + X2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 6.666666667X + X2 = 0 + -0.3333333333
6.666666667X + X2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
6.666666667X + X2 = -0.3333333333

The X term is 6.666666667X.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667X + 11.11111112 + X2 = -0.3333333333 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667X + X2 = -0.3333333333 + 11.11111112

Combine like terms: -0.3333333333 + 11.11111112 = 10.7777777867
11.11111112 + 6.666666667X + X2 = 10.7777777867

Factor a perfect square on the left side:
(X + 3.333333334)(X + 3.333333334) = 10.7777777867

Calculate the square root of the right side: 3.282952602

Break this problem into two subproblems by setting 
(X + 3.333333334) equal to 3.282952602 and -3.282952602.


Subproblem 1
X + 3.333333334 = 3.282952602

Simplifying
X + 3.333333334 = 3.282952602

Reorder the terms:
3.333333334 + X = 3.282952602

Solving
3.333333334 + X = 3.282952602

Solving for variable 'X'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + X = 3.282952602 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + X = 3.282952602 + -3.333333334
X = 3.282952602 + -3.333333334

Combine like terms: 3.282952602 + -3.333333334 = -0.050380732
X = -0.050380732

Simplifying
X = -0.050380732


Subproblem 2
X + 3.333333334 = -3.282952602

Simplifying
X + 3.333333334 = -3.282952602

Reorder the terms:
3.333333334 + X = -3.282952602

Solving
3.333333334 + X = -3.282952602

Solving for variable 'X'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + X = -3.282952602 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + X = -3.282952602 + -3.333333334
X = -3.282952602 + -3.333333334

Combine like terms: -3.282952602 + -3.333333334 = -6.616285936
X = -6.616285936

Simplifying
X = -6.616285936


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {-0.050380732, -6.616285936}

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