a^2+2a-41=0

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Solution for a^2+2a-41=0 equation:


Simplifying
a2 + 2a + -41 = 0

Reorder the terms:
-41 + 2a + a2 = 0

Solving
-41 + 2a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + 2a + 41 + a2 = 0 + 41

Reorder the terms:
-41 + 41 + 2a + a2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + 2a + a2 = 0 + 41
2a + a2 = 0 + 41

Combine like terms: 0 + 41 = 41
2a + a2 = 41

The a term is 2a.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2a + 1 + a2 = 41 + 1

Reorder the terms:
1 + 2a + a2 = 41 + 1

Combine like terms: 41 + 1 = 42
1 + 2a + a2 = 42

Factor a perfect square on the left side:
(a + 1)(a + 1) = 42

Calculate the square root of the right side: 6.480740698

Break this problem into two subproblems by setting 
(a + 1) equal to 6.480740698 and -6.480740698.

Subproblem 1

a + 1 = 6.480740698 Simplifying a + 1 = 6.480740698 Reorder the terms: 1 + a = 6.480740698 Solving 1 + a = 6.480740698 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 6.480740698 + -1 Combine like terms: 1 + -1 = 0 0 + a = 6.480740698 + -1 a = 6.480740698 + -1 Combine like terms: 6.480740698 + -1 = 5.480740698 a = 5.480740698 Simplifying a = 5.480740698

Subproblem 2

a + 1 = -6.480740698 Simplifying a + 1 = -6.480740698 Reorder the terms: 1 + a = -6.480740698 Solving 1 + a = -6.480740698 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -6.480740698 + -1 Combine like terms: 1 + -1 = 0 0 + a = -6.480740698 + -1 a = -6.480740698 + -1 Combine like terms: -6.480740698 + -1 = -7.480740698 a = -7.480740698 Simplifying a = -7.480740698

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.480740698, -7.480740698}

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