a^2+18a-106=-10

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


Simplifying
a2 + 18a + -106 = -10

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

Solving
-106 + 18a + a2 = -10

Solving for variable 'a'.

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '96' to each side of the equation.
-96 + 18a + 96 + a2 = 0 + 96

Reorder the terms:
-96 + 96 + 18a + a2 = 0 + 96

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

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

The a term is 18a.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18a + 81 + a2 = 96 + 81

Reorder the terms:
81 + 18a + a2 = 96 + 81

Combine like terms: 96 + 81 = 177
81 + 18a + a2 = 177

Factor a perfect square on the left side:
(a + 9)(a + 9) = 177

Calculate the square root of the right side: 13.304134696

Break this problem into two subproblems by setting 
(a + 9) equal to 13.304134696 and -13.304134696.

Subproblem 1

a + 9 = 13.304134696 Simplifying a + 9 = 13.304134696 Reorder the terms: 9 + a = 13.304134696 Solving 9 + a = 13.304134696 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + a = 13.304134696 + -9 Combine like terms: 9 + -9 = 0 0 + a = 13.304134696 + -9 a = 13.304134696 + -9 Combine like terms: 13.304134696 + -9 = 4.304134696 a = 4.304134696 Simplifying a = 4.304134696

Subproblem 2

a + 9 = -13.304134696 Simplifying a + 9 = -13.304134696 Reorder the terms: 9 + a = -13.304134696 Solving 9 + a = -13.304134696 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + a = -13.304134696 + -9 Combine like terms: 9 + -9 = 0 0 + a = -13.304134696 + -9 a = -13.304134696 + -9 Combine like terms: -13.304134696 + -9 = -22.304134696 a = -22.304134696 Simplifying a = -22.304134696

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.304134696, -22.304134696}

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