.5a^2+a-12=0

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Solution for .5a^2+a-12=0 equation:


Simplifying
0.5a2 + a + -12 = 0

Reorder the terms:
-12 + a + 0.5a2 = 0

Solving
-12 + a + 0.5a2 = 0

Solving for variable 'a'.

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

Divide each side by '0.5'.
-24 + 2a + a2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 2a + 24 + a2 = 0 + 24

Reorder the terms:
-24 + 24 + 2a + a2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 2a + a2 = 0 + 24
2a + a2 = 0 + 24

Combine like terms: 0 + 24 = 24
2a + a2 = 24

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.
2a + 0.25 + a2 = 24 + 0.25

Reorder the terms:
0.25 + 2a + a2 = 24 + 0.25

Combine like terms: 24 + 0.25 = 24.25
0.25 + 2a + a2 = 24.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(a + 0.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

a + 0.5 = 4.924428901 Simplifying a + 0.5 = 4.924428901 Reorder the terms: 0.5 + a = 4.924428901 Solving 0.5 + a = 4.924428901 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 = 4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.924428901 + -0.5 a = 4.924428901 + -0.5 Combine like terms: 4.924428901 + -0.5 = 4.424428901 a = 4.424428901 Simplifying a = 4.424428901

Subproblem 2

a + 0.5 = -4.924428901 Simplifying a + 0.5 = -4.924428901 Reorder the terms: 0.5 + a = -4.924428901 Solving 0.5 + a = -4.924428901 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 = -4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.924428901 + -0.5 a = -4.924428901 + -0.5 Combine like terms: -4.924428901 + -0.5 = -5.424428901 a = -5.424428901 Simplifying a = -5.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.424428901, -5.424428901}

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