x^2+x=25

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Solution for x^2+x=25 equation: 


Simplifying
x2 + x = 25

Reorder the terms:
x + x2 = 25

Solving
x + x2 = 25

Solving for variable 'x'.

Reorder the terms:
-25 + x + x2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + x + x2 = 0 + 25
x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
x + x2 = 25

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 = 25 + 0.25

Reorder the terms:
0.25 + x + x2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + x + x2 = 25.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + 0.5) equal to 5.024937811 and -5.024937811.


Subproblem 1
x + 0.5 = 5.024937811

Simplifying
x + 0.5 = 5.024937811

Reorder the terms:
0.5 + x = 5.024937811

Solving
0.5 + x = 5.024937811

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 = 5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 5.024937811 + -0.5
x = 5.024937811 + -0.5

Combine like terms: 5.024937811 + -0.5 = 4.524937811
x = 4.524937811

Simplifying
x = 4.524937811


Subproblem 2
x + 0.5 = -5.024937811

Simplifying
x + 0.5 = -5.024937811

Reorder the terms:
0.5 + x = -5.024937811

Solving
0.5 + x = -5.024937811

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 = -5.024937811 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -5.024937811 + -0.5
x = -5.024937811 + -0.5

Combine like terms: -5.024937811 + -0.5 = -5.524937811
x = -5.524937811

Simplifying
x = -5.524937811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.524937811, -5.524937811}

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