w^2+2w-78=0

Solution for w^2+2w-78=0 equation:

Simplifying
w2 + 2w + -78 = 0

Reorder the terms:
-78 + 2w + w2 = 0

Solving
-78 + 2w + w2 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 2w + 78 + w2 = 0 + 78

Reorder the terms:
-78 + 78 + 2w + w2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 2w + w2 = 0 + 78
2w + w2 = 0 + 78

Combine like terms: 0 + 78 = 78
2w + w2 = 78

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

Add '1' to each side of the equation.
2w + 1 + w2 = 78 + 1

Reorder the terms:
1 + 2w + w2 = 78 + 1

Combine like terms: 78 + 1 = 79
1 + 2w + w2 = 79

Factor a perfect square on the left side:
(w + 1)(w + 1) = 79

Calculate the square root of the right side: 8.888194417

Break this problem into two subproblems by setting 
(w + 1) equal to 8.888194417 and -8.888194417.

Subproblem 1
w + 1 = 8.888194417

Simplifying
w + 1 = 8.888194417

Reorder the terms:
1 + w = 8.888194417

Solving
1 + w = 8.888194417

Solving for variable 'w'.

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

Add '-1' to each side of the equation.
1 + -1 + w = 8.888194417 + -1

Combine like terms: 1 + -1 = 0
0 + w = 8.888194417 + -1
w = 8.888194417 + -1

Combine like terms: 8.888194417 + -1 = 7.888194417
w = 7.888194417

Simplifying
w = 7.888194417

Subproblem 2
w + 1 = -8.888194417

Simplifying
w + 1 = -8.888194417

Reorder the terms:
1 + w = -8.888194417

Solving
1 + w = -8.888194417

Solving for variable 'w'.

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

Add '-1' to each side of the equation.
1 + -1 + w = -8.888194417 + -1

Combine like terms: 1 + -1 = 0
0 + w = -8.888194417 + -1
w = -8.888194417 + -1

Combine like terms: -8.888194417 + -1 = -9.888194417
w = -9.888194417

Simplifying
w = -9.888194417

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {7.888194417, -9.888194417}