6x^2+21x+3=0

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Solution for 6x^2+21x+3=0 equation: 


Simplifying
6x2 + 21x + 3 = 0

Reorder the terms:
3 + 21x + 6x2 = 0

Solving
3 + 21x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + 7x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(1 + 7x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + 7x + 2x2 = 0

Solving
1 + 7x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 3.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 3.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 3.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 3.5x + x2 = 0 + -0.5
3.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
3.5x + x2 = -0.5

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = -0.5 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = -0.5 + 3.0625

Combine like terms: -0.5 + 3.0625 = 2.5625
3.0625 + 3.5x + x2 = 2.5625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(x + 1.75) equal to 1.600781059 and -1.600781059.


Subproblem 1
x + 1.75 = 1.600781059

Simplifying
x + 1.75 = 1.600781059

Reorder the terms:
1.75 + x = 1.600781059

Solving
1.75 + x = 1.600781059

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 1.600781059 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 1.600781059 + -1.75
x = 1.600781059 + -1.75

Combine like terms: 1.600781059 + -1.75 = -0.149218941
x = -0.149218941

Simplifying
x = -0.149218941


Subproblem 2
x + 1.75 = -1.600781059

Simplifying
x + 1.75 = -1.600781059

Reorder the terms:
1.75 + x = -1.600781059

Solving
1.75 + x = -1.600781059

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -1.600781059 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -1.600781059 + -1.75
x = -1.600781059 + -1.75

Combine like terms: -1.600781059 + -1.75 = -3.350781059
x = -3.350781059

Simplifying
x = -3.350781059


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.149218941, -3.350781059}
Solution
x = {-0.149218941, -3.350781059}

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