x^2+x=0.78

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Solution for x^2+x=0.78 equation: 


Simplifying
x2 + x = 0.78

Reorder the terms:
x + x2 = 0.78

Solving
x + x2 = 0.78

Solving for variable 'x'.

Reorder the terms:
-0.78 + x + x2 = 0.78 + -0.78

Combine like terms: 0.78 + -0.78 = 0.00
-0.78 + x + x2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '0.78' to each side of the equation.
-0.78 + x + 0.78 + x2 = 0.00 + 0.78

Reorder the terms:
-0.78 + 0.78 + x + x2 = 0.00 + 0.78

Combine like terms: -0.78 + 0.78 = 0.00
0.00 + x + x2 = 0.00 + 0.78
x + x2 = 0.00 + 0.78

Combine like terms: 0.00 + 0.78 = 0.78
x + x2 = 0.78

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 = 0.78 + 0.25

Reorder the terms:
0.25 + x + x2 = 0.78 + 0.25

Combine like terms: 0.78 + 0.25 = 1.03
0.25 + x + x2 = 1.03

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.03

Calculate the square root of the right side: 1.014889157

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.014889157 and -1.014889157.


Subproblem 1
x + 0.5 = 1.014889157

Simplifying
x + 0.5 = 1.014889157

Reorder the terms:
0.5 + x = 1.014889157

Solving
0.5 + x = 1.014889157

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 = 1.014889157 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.014889157 + -0.5
x = 1.014889157 + -0.5

Combine like terms: 1.014889157 + -0.5 = 0.514889157
x = 0.514889157

Simplifying
x = 0.514889157


Subproblem 2
x + 0.5 = -1.014889157

Simplifying
x + 0.5 = -1.014889157

Reorder the terms:
0.5 + x = -1.014889157

Solving
0.5 + x = -1.014889157

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 = -1.014889157 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.014889157 + -0.5
x = -1.014889157 + -0.5

Combine like terms: -1.014889157 + -0.5 = -1.514889157
x = -1.514889157

Simplifying
x = -1.514889157


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.514889157, -1.514889157}

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