(9a^2)+4a=500

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Solution for (9a^2)+4a=500 equation:


Simplifying
(9a2) + 4a = 500

Reorder the terms:
4a + (9a2) = 500

Solving
4a + (9a2) = 500

Solving for variable 'a'.

Reorder the terms:
-500 + 4a + (9a2) = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + 4a + (9a2) = 0

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

Divide each side by '9'.
-55.55555556 + 0.4444444444a + a2 = 0

Move the constant term to the right:

Add '55.55555556' to each side of the equation.
-55.55555556 + 0.4444444444a + 55.55555556 + a2 = 0 + 55.55555556

Reorder the terms:
-55.55555556 + 55.55555556 + 0.4444444444a + a2 = 0 + 55.55555556

Combine like terms: -55.55555556 + 55.55555556 = 0.00000000
0.00000000 + 0.4444444444a + a2 = 0 + 55.55555556
0.4444444444a + a2 = 0 + 55.55555556

Combine like terms: 0 + 55.55555556 = 55.55555556
0.4444444444a + a2 = 55.55555556

The a term is 0.4444444444a.  Take half its coefficient (0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
0.4444444444a + 0.04938271604 + a2 = 55.55555556 + 0.04938271604

Reorder the terms:
0.04938271604 + 0.4444444444a + a2 = 55.55555556 + 0.04938271604

Combine like terms: 55.55555556 + 0.04938271604 = 55.60493827604
0.04938271604 + 0.4444444444a + a2 = 55.60493827604

Factor a perfect square on the left side:
((a) + 0.2222222222)((a) + 0.2222222222) = 55.60493827604

Calculate the square root of the right side: 7.456871883

Break this problem into two subproblems by setting 
((a) + 0.2222222222) equal to 7.456871883 and -7.456871883.

Subproblem 1

(a) + 0.2222222222 = 7.456871883 Simplifying (a) + 0.2222222222 = 7.456871883 a + 0.2222222222 = 7.456871883 Reorder the terms: 0.2222222222 + a = 7.456871883 Solving 0.2222222222 + a = 7.456871883 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + a = 7.456871883 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + a = 7.456871883 + -0.2222222222 a = 7.456871883 + -0.2222222222 Combine like terms: 7.456871883 + -0.2222222222 = 7.2346496608 a = 7.2346496608 Simplifying a = 7.2346496608

Subproblem 2

(a) + 0.2222222222 = -7.456871883 Simplifying (a) + 0.2222222222 = -7.456871883 a + 0.2222222222 = -7.456871883 Reorder the terms: 0.2222222222 + a = -7.456871883 Solving 0.2222222222 + a = -7.456871883 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + a = -7.456871883 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + a = -7.456871883 + -0.2222222222 a = -7.456871883 + -0.2222222222 Combine like terms: -7.456871883 + -0.2222222222 = -7.6790941052 a = -7.6790941052 Simplifying a = -7.6790941052

Solution

The solution to the problem is based on the solutions from the subproblems. a = {7.2346496608, -7.6790941052}

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