x^2-57+x=78

Solution for x^2-57+x=78 equation:

Simplifying
x2 + -57 + x = 78

Reorder the terms:
-57 + x + x2 = 78

Solving
-57 + x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
-57 + -78 + x + x2 = 78 + -78

Combine like terms: -57 + -78 = -135
-135 + x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-135 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '135' to each side of the equation.
-135 + x + 135 + x2 = 0 + 135

Reorder the terms:
-135 + 135 + x + x2 = 0 + 135

Combine like terms: -135 + 135 = 0
0 + x + x2 = 0 + 135
x + x2 = 0 + 135

Combine like terms: 0 + 135 = 135
x + x2 = 135

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 135 + 0.25

Reorder the terms:
0.25 + x + x2 = 135 + 0.25

Combine like terms: 135 + 0.25 = 135.25
0.25 + x + x2 = 135.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 135.25

Calculate the square root of the right side: 11.62970335

Break this problem into two subproblems by setting 
(x + 0.5) equal to 11.62970335 and -11.62970335.

Subproblem 1
x + 0.5 = 11.62970335

Simplifying
x + 0.5 = 11.62970335

Reorder the terms:
0.5 + x = 11.62970335

Solving
0.5 + x = 11.62970335

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 11.62970335 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 11.62970335 + -0.5
x = 11.62970335 + -0.5

Combine like terms: 11.62970335 + -0.5 = 11.12970335
x = 11.12970335

Simplifying
x = 11.12970335

Subproblem 2
x + 0.5 = -11.62970335

Simplifying
x + 0.5 = -11.62970335

Reorder the terms:
0.5 + x = -11.62970335

Solving
0.5 + x = -11.62970335

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -11.62970335 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -11.62970335 + -0.5
x = -11.62970335 + -0.5

Combine like terms: -11.62970335 + -0.5 = -12.12970335
x = -12.12970335

Simplifying
x = -12.12970335

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.12970335, -12.12970335}