a^2+22a+5=0

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Solution for a^2+22a+5=0 equation:


Simplifying
a2 + 22a + 5 = 0

Reorder the terms:
5 + 22a + a2 = 0

Solving
5 + 22a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 22a + -5 + a2 = 0 + -5

Reorder the terms:
5 + -5 + 22a + a2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 22a + a2 = 0 + -5
22a + a2 = 0 + -5

Combine like terms: 0 + -5 = -5
22a + a2 = -5

The a term is 22a.  Take half its coefficient (11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
22a + 121 + a2 = -5 + 121

Reorder the terms:
121 + 22a + a2 = -5 + 121

Combine like terms: -5 + 121 = 116
121 + 22a + a2 = 116

Factor a perfect square on the left side:
(a + 11)(a + 11) = 116

Calculate the square root of the right side: 10.770329614

Break this problem into two subproblems by setting 
(a + 11) equal to 10.770329614 and -10.770329614.

Subproblem 1

a + 11 = 10.770329614 Simplifying a + 11 = 10.770329614 Reorder the terms: 11 + a = 10.770329614 Solving 11 + a = 10.770329614 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-11' to each side of the equation. 11 + -11 + a = 10.770329614 + -11 Combine like terms: 11 + -11 = 0 0 + a = 10.770329614 + -11 a = 10.770329614 + -11 Combine like terms: 10.770329614 + -11 = -0.229670386 a = -0.229670386 Simplifying a = -0.229670386

Subproblem 2

a + 11 = -10.770329614 Simplifying a + 11 = -10.770329614 Reorder the terms: 11 + a = -10.770329614 Solving 11 + a = -10.770329614 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-11' to each side of the equation. 11 + -11 + a = -10.770329614 + -11 Combine like terms: 11 + -11 = 0 0 + a = -10.770329614 + -11 a = -10.770329614 + -11 Combine like terms: -10.770329614 + -11 = -21.770329614 a = -21.770329614 Simplifying a = -21.770329614

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.229670386, -21.770329614}

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