0=x^2+11x-78

Solution for 0=x^2+11x-78 equation:

Simplifying
0 = x2 + 11x + -78

Reorder the terms:
0 = -78 + 11x + x2

Solving
0 = -78 + 11x + x2

Solving for variable 'x'.

Combine like terms: 0 + 78 = 78
78 + -11x + -1x2 = -78 + 11x + x2 + 78 + -11x + -1x2

Reorder the terms:
78 + -11x + -1x2 = -78 + 78 + 11x + -11x + x2 + -1x2

Combine like terms: -78 + 78 = 0
78 + -11x + -1x2 = 0 + 11x + -11x + x2 + -1x2
78 + -11x + -1x2 = 11x + -11x + x2 + -1x2

Combine like terms: 11x + -11x = 0
78 + -11x + -1x2 = 0 + x2 + -1x2
78 + -11x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
78 + -11x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-78 + 11x + x2 = 0

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 11x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + 11x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 11x + x2 = 0 + 78
11x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
11x + x2 = 78

The x term is 11x.  Take half its coefficient (5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
11x + 30.25 + x2 = 78 + 30.25

Reorder the terms:
30.25 + 11x + x2 = 78 + 30.25

Combine like terms: 78 + 30.25 = 108.25
30.25 + 11x + x2 = 108.25

Factor a perfect square on the left side:
(x + 5.5)(x + 5.5) = 108.25

Calculate the square root of the right side: 10.404326023

Break this problem into two subproblems by setting 
(x + 5.5) equal to 10.404326023 and -10.404326023.

Subproblem 1
x + 5.5 = 10.404326023

Simplifying
x + 5.5 = 10.404326023

Reorder the terms:
5.5 + x = 10.404326023

Solving
5.5 + x = 10.404326023

Solving for variable 'x'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + x = 10.404326023 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + x = 10.404326023 + -5.5
x = 10.404326023 + -5.5

Combine like terms: 10.404326023 + -5.5 = 4.904326023
x = 4.904326023

Simplifying
x = 4.904326023

Subproblem 2
x + 5.5 = -10.404326023

Simplifying
x + 5.5 = -10.404326023

Reorder the terms:
5.5 + x = -10.404326023

Solving
5.5 + x = -10.404326023

Solving for variable 'x'.

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

Add '-5.5' to each side of the equation.
5.5 + -5.5 + x = -10.404326023 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + x = -10.404326023 + -5.5
x = -10.404326023 + -5.5

Combine like terms: -10.404326023 + -5.5 = -15.904326023
x = -15.904326023

Simplifying
x = -15.904326023

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.904326023, -15.904326023}