3a^2+16a+21=12

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Solution for 3a^2+16a+21=12 equation: 


Simplifying
3a2 + 16a + 21 = 12

Reorder the terms:
21 + 16a + 3a2 = 12

Solving
21 + 16a + 3a2 = 12

Solving for variable 'a'.

Reorder the terms:
21 + -12 + 16a + 3a2 = 12 + -12

Combine like terms: 21 + -12 = 9
9 + 16a + 3a2 = 12 + -12

Combine like terms: 12 + -12 = 0
9 + 16a + 3a2 = 0

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

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

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5.333333333a + -3 + a2 = 0 + -3

Reorder the terms:
3 + -3 + 5.333333333a + a2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5.333333333a + a2 = 0 + -3
5.333333333a + a2 = 0 + -3

Combine like terms: 0 + -3 = -3
5.333333333a + a2 = -3

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 = -3 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333a + a2 = -3 + 7.111111113

Combine like terms: -3 + 7.111111113 = 4.111111113
7.111111113 + 5.333333333a + a2 = 4.111111113

Factor a perfect square on the left side:
(a + 2.666666667)(a + 2.666666667) = 4.111111113

Calculate the square root of the right side: 2.027587511

Break this problem into two subproblems by setting 
(a + 2.666666667) equal to 2.027587511 and -2.027587511.


Subproblem 1
a + 2.666666667 = 2.027587511

Simplifying
a + 2.666666667 = 2.027587511

Reorder the terms:
2.666666667 + a = 2.027587511

Solving
2.666666667 + a = 2.027587511

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 = 2.027587511 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + a = 2.027587511 + -2.666666667
a = 2.027587511 + -2.666666667

Combine like terms: 2.027587511 + -2.666666667 = -0.639079156
a = -0.639079156

Simplifying
a = -0.639079156


Subproblem 2
a + 2.666666667 = -2.027587511

Simplifying
a + 2.666666667 = -2.027587511

Reorder the terms:
2.666666667 + a = -2.027587511

Solving
2.666666667 + a = -2.027587511

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 = -2.027587511 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + a = -2.027587511 + -2.666666667
a = -2.027587511 + -2.666666667

Combine like terms: -2.027587511 + -2.666666667 = -4.694254178
a = -4.694254178

Simplifying
a = -4.694254178


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.639079156, -4.694254178}

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