a^2+10a+22=0

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Solution for a^2+10a+22=0 equation: 


Simplifying
a2 + 10a + 22 = 0

Reorder the terms:
22 + 10a + a2 = 0

Solving
22 + 10a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + 10a + -22 + a2 = 0 + -22

Reorder the terms:
22 + -22 + 10a + a2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 10a + a2 = 0 + -22
10a + a2 = 0 + -22

Combine like terms: 0 + -22 = -22
10a + a2 = -22

The a term is 10a.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10a + 25 + a2 = -22 + 25

Reorder the terms:
25 + 10a + a2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + 10a + a2 = 3

Factor a perfect square on the left side:
(a + 5)(a + 5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(a + 5) equal to 1.732050808 and -1.732050808.


Subproblem 1
a + 5 = 1.732050808

Simplifying
a + 5 = 1.732050808

Reorder the terms:
5 + a = 1.732050808

Solving
5 + a = 1.732050808

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-5' to each side of the equation.
5 + -5 + a = 1.732050808 + -5

Combine like terms: 5 + -5 = 0
0 + a = 1.732050808 + -5
a = 1.732050808 + -5

Combine like terms: 1.732050808 + -5 = -3.267949192
a = -3.267949192

Simplifying
a = -3.267949192


Subproblem 2
a + 5 = -1.732050808

Simplifying
a + 5 = -1.732050808

Reorder the terms:
5 + a = -1.732050808

Solving
5 + a = -1.732050808

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-5' to each side of the equation.
5 + -5 + a = -1.732050808 + -5

Combine like terms: 5 + -5 = 0
0 + a = -1.732050808 + -5
a = -1.732050808 + -5

Combine like terms: -1.732050808 + -5 = -6.732050808
a = -6.732050808

Simplifying
a = -6.732050808


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-3.267949192, -6.732050808}

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