x^2+x=19.25

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Solution for x^2+x=19.25 equation: 


Simplifying
x2 + x = 19.25

Reorder the terms:
x + x2 = 19.25

Solving
x + x2 = 19.25

Solving for variable 'x'.

Reorder the terms:
-19.25 + x + x2 = 19.25 + -19.25

Combine like terms: 19.25 + -19.25 = 0.00
-19.25 + x + x2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '19.25' to each side of the equation.
-19.25 + x + 19.25 + x2 = 0.00 + 19.25

Reorder the terms:
-19.25 + 19.25 + x + x2 = 0.00 + 19.25

Combine like terms: -19.25 + 19.25 = 0.00
0.00 + x + x2 = 0.00 + 19.25
x + x2 = 0.00 + 19.25

Combine like terms: 0.00 + 19.25 = 19.25
x + x2 = 19.25

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 = 19.25 + 0.25

Reorder the terms:
0.25 + x + x2 = 19.25 + 0.25

Combine like terms: 19.25 + 0.25 = 19.5
0.25 + x + x2 = 19.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 19.5

Calculate the square root of the right side: 4.415880433

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.415880433 and -4.415880433.


Subproblem 1
x + 0.5 = 4.415880433

Simplifying
x + 0.5 = 4.415880433

Reorder the terms:
0.5 + x = 4.415880433

Solving
0.5 + x = 4.415880433

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 = 4.415880433 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.415880433 + -0.5
x = 4.415880433 + -0.5

Combine like terms: 4.415880433 + -0.5 = 3.915880433
x = 3.915880433

Simplifying
x = 3.915880433


Subproblem 2
x + 0.5 = -4.415880433

Simplifying
x + 0.5 = -4.415880433

Reorder the terms:
0.5 + x = -4.415880433

Solving
0.5 + x = -4.415880433

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 = -4.415880433 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.415880433 + -0.5
x = -4.415880433 + -0.5

Combine like terms: -4.415880433 + -0.5 = -4.915880433
x = -4.915880433

Simplifying
x = -4.915880433


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.915880433, -4.915880433}

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