-x^2+84x-16=0

Solution for -x^2+84x-16=0 equation:

Simplifying
-1x2 + 84x + -16 = 0

Reorder the terms:
-16 + 84x + -1x2 = 0

Solving
-16 + 84x + -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'.
16 + -84x + x2 = 0

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -84x + -16 + x2 = 0 + -16

Reorder the terms:
16 + -16 + -84x + x2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -84x + x2 = 0 + -16
-84x + x2 = 0 + -16

Combine like terms: 0 + -16 = -16
-84x + x2 = -16

The x term is -84x.  Take half its coefficient (-42).
Square it (1764) and add it to both sides.

Add '1764' to each side of the equation.
-84x + 1764 + x2 = -16 + 1764

Reorder the terms:
1764 + -84x + x2 = -16 + 1764

Combine like terms: -16 + 1764 = 1748
1764 + -84x + x2 = 1748

Factor a perfect square on the left side:
(x + -42)(x + -42) = 1748

Calculate the square root of the right side: 41.809089921

Break this problem into two subproblems by setting 
(x + -42) equal to 41.809089921 and -41.809089921.

Subproblem 1
x + -42 = 41.809089921

Simplifying
x + -42 = 41.809089921

Reorder the terms:
-42 + x = 41.809089921

Solving
-42 + x = 41.809089921

Solving for variable 'x'.

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

Add '42' to each side of the equation.
-42 + 42 + x = 41.809089921 + 42

Combine like terms: -42 + 42 = 0
0 + x = 41.809089921 + 42
x = 41.809089921 + 42

Combine like terms: 41.809089921 + 42 = 83.809089921
x = 83.809089921

Simplifying
x = 83.809089921

Subproblem 2
x + -42 = -41.809089921

Simplifying
x + -42 = -41.809089921

Reorder the terms:
-42 + x = -41.809089921

Solving
-42 + x = -41.809089921

Solving for variable 'x'.

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

Add '42' to each side of the equation.
-42 + 42 + x = -41.809089921 + 42

Combine like terms: -42 + 42 = 0
0 + x = -41.809089921 + 42
x = -41.809089921 + 42

Combine like terms: -41.809089921 + 42 = 0.190910079
x = 0.190910079

Simplifying
x = 0.190910079

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {83.809089921, 0.190910079}