-x^2+19x+78=0

Solution for -x^2+19x+78=0 equation:

Simplifying
-1x2 + 19x + 78 = 0

Reorder the terms:
78 + 19x + -1x2 = 0

Solving
78 + 19x + -1x2 = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
-78 + -19x + x2 = 0

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + -19x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + -19x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -19x + x2 = 0 + 78
-19x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
-19x + x2 = 78

The x term is -19x.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19x + 90.25 + x2 = 78 + 90.25

Reorder the terms:
90.25 + -19x + x2 = 78 + 90.25

Combine like terms: 78 + 90.25 = 168.25
90.25 + -19x + x2 = 168.25

Factor a perfect square on the left side:
(x + -9.5)(x + -9.5) = 168.25

Calculate the square root of the right side: 12.971121771

Break this problem into two subproblems by setting 
(x + -9.5) equal to 12.971121771 and -12.971121771.

Subproblem 1
x + -9.5 = 12.971121771

Simplifying
x + -9.5 = 12.971121771

Reorder the terms:
-9.5 + x = 12.971121771

Solving
-9.5 + x = 12.971121771

Solving for variable 'x'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + x = 12.971121771 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + x = 12.971121771 + 9.5
x = 12.971121771 + 9.5

Combine like terms: 12.971121771 + 9.5 = 22.471121771
x = 22.471121771

Simplifying
x = 22.471121771

Subproblem 2
x + -9.5 = -12.971121771

Simplifying
x + -9.5 = -12.971121771

Reorder the terms:
-9.5 + x = -12.971121771

Solving
-9.5 + x = -12.971121771

Solving for variable 'x'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + x = -12.971121771 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + x = -12.971121771 + 9.5
x = -12.971121771 + 9.5

Combine like terms: -12.971121771 + 9.5 = -3.471121771
x = -3.471121771

Simplifying
x = -3.471121771

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {22.471121771, -3.471121771}