4a^2+9a+3=0

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Solution for 4a^2+9a+3=0 equation: 


Simplifying
4a2 + 9a + 3 = 0

Reorder the terms:
3 + 9a + 4a2 = 0

Solving
3 + 9a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
0.75 + 2.25a + a2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.25a + -0.75 + a2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.25a + a2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.25a + a2 = 0 + -0.75
2.25a + a2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.25a + a2 = -0.75

The a term is 2.25a.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25a + 1.265625 + a2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25a + a2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + 2.25a + a2 = 0.515625

Factor a perfect square on the left side:
(a + 1.125)(a + 1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(a + 1.125) equal to 0.718070331 and -0.718070331.


Subproblem 1
a + 1.125 = 0.718070331

Simplifying
a + 1.125 = 0.718070331

Reorder the terms:
1.125 + a = 0.718070331

Solving
1.125 + a = 0.718070331

Solving for variable 'a'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + a = 0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + a = 0.718070331 + -1.125
a = 0.718070331 + -1.125

Combine like terms: 0.718070331 + -1.125 = -0.406929669
a = -0.406929669

Simplifying
a = -0.406929669


Subproblem 2
a + 1.125 = -0.718070331

Simplifying
a + 1.125 = -0.718070331

Reorder the terms:
1.125 + a = -0.718070331

Solving
1.125 + a = -0.718070331

Solving for variable 'a'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + a = -0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + a = -0.718070331 + -1.125
a = -0.718070331 + -1.125

Combine like terms: -0.718070331 + -1.125 = -1.843070331
a = -1.843070331

Simplifying
a = -1.843070331


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.406929669, -1.843070331}

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