3*X^2+5X+1=0

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Solution for 3*X^2+5X+1=0 equation: 


Simplifying
3X2 + 5X + 1 = 0

Reorder the terms:
1 + 5X + 3X2 = 0

Solving
1 + 5X + 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 + 1.666666667X + X2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 1.666666667X + -0.3333333333 + X2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 1.666666667X + X2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 1.666666667X + X2 = 0 + -0.3333333333
1.666666667X + X2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
1.666666667X + X2 = -0.3333333333

The X term is 1.666666667X.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667X + 0.6944444447 + X2 = -0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667X + X2 = -0.3333333333 + 0.6944444447

Combine like terms: -0.3333333333 + 0.6944444447 = 0.3611111114
0.6944444447 + 1.666666667X + X2 = 0.3611111114

Factor a perfect square on the left side:
(X + 0.8333333335)(X + 0.8333333335) = 0.3611111114

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(X + 0.8333333335) equal to 0.600925213 and -0.600925213.


Subproblem 1
X + 0.8333333335 = 0.600925213

Simplifying
X + 0.8333333335 = 0.600925213

Reorder the terms:
0.8333333335 + X = 0.600925213

Solving
0.8333333335 + X = 0.600925213

Solving for variable 'X'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + X = 0.600925213 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + X = 0.600925213 + -0.8333333335
X = 0.600925213 + -0.8333333335

Combine like terms: 0.600925213 + -0.8333333335 = -0.2324081205
X = -0.2324081205

Simplifying
X = -0.2324081205


Subproblem 2
X + 0.8333333335 = -0.600925213

Simplifying
X + 0.8333333335 = -0.600925213

Reorder the terms:
0.8333333335 + X = -0.600925213

Solving
0.8333333335 + X = -0.600925213

Solving for variable 'X'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + X = -0.600925213 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + X = -0.600925213 + -0.8333333335
X = -0.600925213 + -0.8333333335

Combine like terms: -0.600925213 + -0.8333333335 = -1.4342585465
X = -1.4342585465

Simplifying
X = -1.4342585465


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {-0.2324081205, -1.4342585465}

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