a^2+40a+22=0

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Solution for a^2+40a+22=0 equation:


Simplifying
a2 + 40a + 22 = 0

Reorder the terms:
22 + 40a + a2 = 0

Solving
22 + 40a + 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 + 40a + -22 + a2 = 0 + -22

Reorder the terms:
22 + -22 + 40a + a2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + 40a + a2 = 0 + -22
40a + a2 = 0 + -22

Combine like terms: 0 + -22 = -22
40a + a2 = -22

The a term is 40a.  Take half its coefficient (20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
40a + 400 + a2 = -22 + 400

Reorder the terms:
400 + 40a + a2 = -22 + 400

Combine like terms: -22 + 400 = 378
400 + 40a + a2 = 378

Factor a perfect square on the left side:
(a + 20)(a + 20) = 378

Calculate the square root of the right side: 19.442222095

Break this problem into two subproblems by setting 
(a + 20) equal to 19.442222095 and -19.442222095.

Subproblem 1

a + 20 = 19.442222095 Simplifying a + 20 = 19.442222095 Reorder the terms: 20 + a = 19.442222095 Solving 20 + a = 19.442222095 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + a = 19.442222095 + -20 Combine like terms: 20 + -20 = 0 0 + a = 19.442222095 + -20 a = 19.442222095 + -20 Combine like terms: 19.442222095 + -20 = -0.557777905 a = -0.557777905 Simplifying a = -0.557777905

Subproblem 2

a + 20 = -19.442222095 Simplifying a + 20 = -19.442222095 Reorder the terms: 20 + a = -19.442222095 Solving 20 + a = -19.442222095 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-20' to each side of the equation. 20 + -20 + a = -19.442222095 + -20 Combine like terms: 20 + -20 = 0 0 + a = -19.442222095 + -20 a = -19.442222095 + -20 Combine like terms: -19.442222095 + -20 = -39.442222095 a = -39.442222095 Simplifying a = -39.442222095

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.557777905, -39.442222095}

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