9a^2+a-3=0

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


Simplifying
9a2 + a + -3 = 0

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

Solving
-3 + a + 9a2 = 0

Solving for variable 'a'.

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

Divide each side by '9'.
-0.3333333333 + 0.1111111111a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.1111111111a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.1111111111a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.1111111111a + a2 = 0 + 0.3333333333
0.1111111111a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.1111111111a + a2 = 0.3333333333

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1111111111a + 0.25 + a2 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + 0.1111111111a + a2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + 0.1111111111a + a2 = 0.5833333333

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(a + 0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

a + 0.5 = 0.763762616 Simplifying a + 0.5 = 0.763762616 Reorder the terms: 0.5 + a = 0.763762616 Solving 0.5 + a = 0.763762616 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.763762616 + -0.5 a = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 a = 0.263762616 Simplifying a = 0.263762616

Subproblem 2

a + 0.5 = -0.763762616 Simplifying a + 0.5 = -0.763762616 Reorder the terms: 0.5 + a = -0.763762616 Solving 0.5 + a = -0.763762616 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.763762616 + -0.5 a = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 a = -1.263762616 Simplifying a = -1.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.263762616, -1.263762616}

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