a+18a^2-106=-10

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Solution for a+18a^2-106=-10 equation:


Simplifying
a + 18a2 + -106 = -10

Reorder the terms:
-106 + a + 18a2 = -10

Solving
-106 + a + 18a2 = -10

Solving for variable 'a'.

Reorder the terms:
-106 + 10 + a + 18a2 = -10 + 10

Combine like terms: -106 + 10 = -96
-96 + a + 18a2 = -10 + 10

Combine like terms: -10 + 10 = 0
-96 + a + 18a2 = 0

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

Divide each side by '18'.
-5.333333333 + 0.05555555556a + a2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + 0.05555555556a + 5.333333333 + a2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + 0.05555555556a + a2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + 0.05555555556a + a2 = 0 + 5.333333333
0.05555555556a + a2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
0.05555555556a + a2 = 5.333333333

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.
0.05555555556a + 0.25 + a2 = 5.333333333 + 0.25

Reorder the terms:
0.25 + 0.05555555556a + a2 = 5.333333333 + 0.25

Combine like terms: 5.333333333 + 0.25 = 5.583333333
0.25 + 0.05555555556a + a2 = 5.583333333

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 5.583333333

Calculate the square root of the right side: 2.362907813

Break this problem into two subproblems by setting 
(a + 0.5) equal to 2.362907813 and -2.362907813.

Subproblem 1

a + 0.5 = 2.362907813 Simplifying a + 0.5 = 2.362907813 Reorder the terms: 0.5 + a = 2.362907813 Solving 0.5 + a = 2.362907813 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 = 2.362907813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 2.362907813 + -0.5 a = 2.362907813 + -0.5 Combine like terms: 2.362907813 + -0.5 = 1.862907813 a = 1.862907813 Simplifying a = 1.862907813

Subproblem 2

a + 0.5 = -2.362907813 Simplifying a + 0.5 = -2.362907813 Reorder the terms: 0.5 + a = -2.362907813 Solving 0.5 + a = -2.362907813 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 = -2.362907813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -2.362907813 + -0.5 a = -2.362907813 + -0.5 Combine like terms: -2.362907813 + -0.5 = -2.862907813 a = -2.862907813 Simplifying a = -2.862907813

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.862907813, -2.862907813}

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