3a^2+12a-1=0

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Solution for 3a^2+12a-1=0 equation:


Simplifying
3a2 + 12a + -1 = 0

Reorder the terms:
-1 + 12a + 3a2 = 0

Solving
-1 + 12a + 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'.
-0.3333333333 + 4a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 4a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 4a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 4a + a2 = 0 + 0.3333333333
4a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
4a + a2 = 0.3333333333

The a term is 4a.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4a + 4 + a2 = 0.3333333333 + 4

Reorder the terms:
4 + 4a + a2 = 0.3333333333 + 4

Combine like terms: 0.3333333333 + 4 = 4.3333333333
4 + 4a + a2 = 4.3333333333

Factor a perfect square on the left side:
(a + 2)(a + 2) = 4.3333333333

Calculate the square root of the right side: 2.081665999

Break this problem into two subproblems by setting 
(a + 2) equal to 2.081665999 and -2.081665999.

Subproblem 1

a + 2 = 2.081665999 Simplifying a + 2 = 2.081665999 Reorder the terms: 2 + a = 2.081665999 Solving 2 + a = 2.081665999 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = 2.081665999 + -2 Combine like terms: 2 + -2 = 0 0 + a = 2.081665999 + -2 a = 2.081665999 + -2 Combine like terms: 2.081665999 + -2 = 0.081665999 a = 0.081665999 Simplifying a = 0.081665999

Subproblem 2

a + 2 = -2.081665999 Simplifying a + 2 = -2.081665999 Reorder the terms: 2 + a = -2.081665999 Solving 2 + a = -2.081665999 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + a = -2.081665999 + -2 Combine like terms: 2 + -2 = 0 0 + a = -2.081665999 + -2 a = -2.081665999 + -2 Combine like terms: -2.081665999 + -2 = -4.081665999 a = -4.081665999 Simplifying a = -4.081665999

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.081665999, -4.081665999}

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