3a^2+16a+3=0

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Solution for 3a^2+16a+3=0 equation: 


Simplifying
3a2 + 16a + 3 = 0

Reorder the terms:
3 + 16a + 3a2 = 0

Solving
3 + 16a + 3a2 = 0

Solving for variable 'a'.

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

Divide each side by '3'.
1 + 5.333333333a + a2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 5.333333333a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 5.333333333a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 5.333333333a + a2 = 0 + -1
5.333333333a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
5.333333333a + a2 = -1

The a term is 5.333333333a.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333a + 7.111111113 + a2 = -1 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333a + a2 = -1 + 7.111111113

Combine like terms: -1 + 7.111111113 = 6.111111113
7.111111113 + 5.333333333a + a2 = 6.111111113

Factor a perfect square on the left side:
(a + 2.666666667)(a + 2.666666667) = 6.111111113

Calculate the square root of the right side: 2.472066163

Break this problem into two subproblems by setting 
(a + 2.666666667) equal to 2.472066163 and -2.472066163.


Subproblem 1
a + 2.666666667 = 2.472066163

Simplifying
a + 2.666666667 = 2.472066163

Reorder the terms:
2.666666667 + a = 2.472066163

Solving
2.666666667 + a = 2.472066163

Solving for variable 'a'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + a = 2.472066163 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + a = 2.472066163 + -2.666666667
a = 2.472066163 + -2.666666667

Combine like terms: 2.472066163 + -2.666666667 = -0.194600504
a = -0.194600504

Simplifying
a = -0.194600504


Subproblem 2
a + 2.666666667 = -2.472066163

Simplifying
a + 2.666666667 = -2.472066163

Reorder the terms:
2.666666667 + a = -2.472066163

Solving
2.666666667 + a = -2.472066163

Solving for variable 'a'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + a = -2.472066163 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + a = -2.472066163 + -2.666666667
a = -2.472066163 + -2.666666667

Combine like terms: -2.472066163 + -2.666666667 = -5.13873283
a = -5.13873283

Simplifying
a = -5.13873283


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.194600504, -5.13873283}

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