0.2a^2+12a+0.9=0

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Solution for 0.2a^2+12a+0.9=0 equation:


Simplifying
0.2a2 + 12a + 0.9 = 0

Reorder the terms:
0.9 + 12a + 0.2a2 = 0

Solving
0.9 + 12a + 0.2a2 = 0

Solving for variable 'a'.

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

Divide each side by '0.2'.
4.5 + 60a + a2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 60a + -4.5 + a2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 60a + a2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 60a + a2 = 0 + -4.5
60a + a2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
60a + a2 = -4.5

The a term is 60a.  Take half its coefficient (30).
Square it (900) and add it to both sides.

Add '900' to each side of the equation.
60a + 900 + a2 = -4.5 + 900

Reorder the terms:
900 + 60a + a2 = -4.5 + 900

Combine like terms: -4.5 + 900 = 895.5
900 + 60a + a2 = 895.5

Factor a perfect square on the left side:
(a + 30)(a + 30) = 895.5

Calculate the square root of the right side: 29.924906015

Break this problem into two subproblems by setting 
(a + 30) equal to 29.924906015 and -29.924906015.

Subproblem 1

a + 30 = 29.924906015 Simplifying a + 30 = 29.924906015 Reorder the terms: 30 + a = 29.924906015 Solving 30 + a = 29.924906015 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-30' to each side of the equation. 30 + -30 + a = 29.924906015 + -30 Combine like terms: 30 + -30 = 0 0 + a = 29.924906015 + -30 a = 29.924906015 + -30 Combine like terms: 29.924906015 + -30 = -0.075093985 a = -0.075093985 Simplifying a = -0.075093985

Subproblem 2

a + 30 = -29.924906015 Simplifying a + 30 = -29.924906015 Reorder the terms: 30 + a = -29.924906015 Solving 30 + a = -29.924906015 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-30' to each side of the equation. 30 + -30 + a = -29.924906015 + -30 Combine like terms: 30 + -30 = 0 0 + a = -29.924906015 + -30 a = -29.924906015 + -30 Combine like terms: -29.924906015 + -30 = -59.924906015 a = -59.924906015 Simplifying a = -59.924906015

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.075093985, -59.924906015}

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