6*x^2+15*x+3=0

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Solution for 6*x^2+15*x+3=0 equation: 


Simplifying
6x2 + 15x + 3 = 0

Reorder the terms:
3 + 15x + 6x2 = 0

Solving
3 + 15x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + 5x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(1 + 5x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + 5x + 2x2 = 0

Solving
1 + 5x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.5x + x2 = 0 + -0.5
2.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.5x + x2 = -0.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + 2.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 1.25) equal to 1.030776406 and -1.030776406.


Subproblem 1
x + 1.25 = 1.030776406

Simplifying
x + 1.25 = 1.030776406

Reorder the terms:
1.25 + x = 1.030776406

Solving
1.25 + x = 1.030776406

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = 1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 1.030776406 + -1.25
x = 1.030776406 + -1.25

Combine like terms: 1.030776406 + -1.25 = -0.219223594
x = -0.219223594

Simplifying
x = -0.219223594


Subproblem 2
x + 1.25 = -1.030776406

Simplifying
x + 1.25 = -1.030776406

Reorder the terms:
1.25 + x = -1.030776406

Solving
1.25 + x = -1.030776406

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = -1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -1.030776406 + -1.25
x = -1.030776406 + -1.25

Combine like terms: -1.030776406 + -1.25 = -2.280776406
x = -2.280776406

Simplifying
x = -2.280776406


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.219223594, -2.280776406}
Solution
x = {-0.219223594, -2.280776406}

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