a^2-31-2a=-6-3a^2-a

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Solution for a^2-31-2a=-6-3a^2-a equation:


Simplifying
a2 + -31 + -2a = -6 + -3a2 + -1a

Reorder the terms:
-31 + -2a + a2 = -6 + -3a2 + -1a

Reorder the terms:
-31 + -2a + a2 = -6 + -1a + -3a2

Solving
-31 + -2a + a2 = -6 + -1a + -3a2

Solving for variable 'a'.

Reorder the terms:
-31 + 6 + -2a + a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2

Combine like terms: -31 + 6 = -25
-25 + -2a + a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2

Combine like terms: -2a + a = -1a
-25 + -1a + a2 + 3a2 = -6 + -1a + -3a2 + 6 + a + 3a2

Combine like terms: a2 + 3a2 = 4a2
-25 + -1a + 4a2 = -6 + -1a + -3a2 + 6 + a + 3a2

Reorder the terms:
-25 + -1a + 4a2 = -6 + 6 + -1a + a + -3a2 + 3a2

Combine like terms: -6 + 6 = 0
-25 + -1a + 4a2 = 0 + -1a + a + -3a2 + 3a2
-25 + -1a + 4a2 = -1a + a + -3a2 + 3a2

Combine like terms: -1a + a = 0
-25 + -1a + 4a2 = 0 + -3a2 + 3a2
-25 + -1a + 4a2 = -3a2 + 3a2

Combine like terms: -3a2 + 3a2 = 0
-25 + -1a + 4a2 = 0

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

Divide each side by '4'.
-6.25 + -0.25a + a2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + -0.25a + 6.25 + a2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + -0.25a + a2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + -0.25a + a2 = 0 + 6.25
-0.25a + a2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
-0.25a + a2 = 6.25

The a term is -0.25a.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25a + 0.015625 + a2 = 6.25 + 0.015625

Reorder the terms:
0.015625 + -0.25a + a2 = 6.25 + 0.015625

Combine like terms: 6.25 + 0.015625 = 6.265625
0.015625 + -0.25a + a2 = 6.265625

Factor a perfect square on the left side:
(a + -0.125)(a + -0.125) = 6.265625

Calculate the square root of the right side: 2.503123049

Break this problem into two subproblems by setting 
(a + -0.125) equal to 2.503123049 and -2.503123049.

Subproblem 1

a + -0.125 = 2.503123049 Simplifying a + -0.125 = 2.503123049 Reorder the terms: -0.125 + a = 2.503123049 Solving -0.125 + a = 2.503123049 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = 2.503123049 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = 2.503123049 + 0.125 a = 2.503123049 + 0.125 Combine like terms: 2.503123049 + 0.125 = 2.628123049 a = 2.628123049 Simplifying a = 2.628123049

Subproblem 2

a + -0.125 = -2.503123049 Simplifying a + -0.125 = -2.503123049 Reorder the terms: -0.125 + a = -2.503123049 Solving -0.125 + a = -2.503123049 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + a = -2.503123049 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + a = -2.503123049 + 0.125 a = -2.503123049 + 0.125 Combine like terms: -2.503123049 + 0.125 = -2.378123049 a = -2.378123049 Simplifying a = -2.378123049

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.628123049, -2.378123049}

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