x+x*x=7.8

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Solution for x+x*x=7.8 equation: 


Simplifying
x + x * x = 7.8

Multiply x * x
x + x2 = 7.8

Solving
x + x2 = 7.8

Solving for variable 'x'.

Reorder the terms:
-7.8 + x + x2 = 7.8 + -7.8

Combine like terms: 7.8 + -7.8 = 0.0
-7.8 + x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '7.8' to each side of the equation.
-7.8 + x + 7.8 + x2 = 0.0 + 7.8

Reorder the terms:
-7.8 + 7.8 + x + x2 = 0.0 + 7.8

Combine like terms: -7.8 + 7.8 = 0.0
0.0 + x + x2 = 0.0 + 7.8
x + x2 = 0.0 + 7.8

Combine like terms: 0.0 + 7.8 = 7.8
x + x2 = 7.8

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 = 7.8 + 0.25

Reorder the terms:
0.25 + x + x2 = 7.8 + 0.25

Combine like terms: 7.8 + 0.25 = 8.05
0.25 + x + x2 = 8.05

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.05

Calculate the square root of the right side: 2.837252192

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.837252192 and -2.837252192.


Subproblem 1
x + 0.5 = 2.837252192

Simplifying
x + 0.5 = 2.837252192

Reorder the terms:
0.5 + x = 2.837252192

Solving
0.5 + x = 2.837252192

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 = 2.837252192 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.837252192 + -0.5
x = 2.837252192 + -0.5

Combine like terms: 2.837252192 + -0.5 = 2.337252192
x = 2.337252192

Simplifying
x = 2.337252192


Subproblem 2
x + 0.5 = -2.837252192

Simplifying
x + 0.5 = -2.837252192

Reorder the terms:
0.5 + x = -2.837252192

Solving
0.5 + x = -2.837252192

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 = -2.837252192 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.837252192 + -0.5
x = -2.837252192 + -0.5

Combine like terms: -2.837252192 + -0.5 = -3.337252192
x = -3.337252192

Simplifying
x = -3.337252192


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.337252192, -3.337252192}

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