=(4x+1)+(x^2-5)

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Solution for =(4x+1)+(x^2-5) equation: 


Simplifying
0 = (4x + 1) + (x2 + -5)

Reorder the terms:
0 = (1 + 4x) + (x2 + -5)

Remove parenthesis around (1 + 4x)
0 = 1 + 4x + (x2 + -5)

Reorder the terms:
0 = 1 + 4x + (-5 + x2)

Remove parenthesis around (-5 + x2)
0 = 1 + 4x + -5 + x2

Reorder the terms:
0 = 1 + -5 + 4x + x2

Combine like terms: 1 + -5 = -4
0 = -4 + 4x + x2

Solving
0 = -4 + 4x + x2

Solving for variable 'x'.

Combine like terms: 0 + 4 = 4
4 + -4x + -1x2 = -4 + 4x + x2 + 4 + -4x + -1x2

Reorder the terms:
4 + -4x + -1x2 = -4 + 4 + 4x + -4x + x2 + -1x2

Combine like terms: -4 + 4 = 0
4 + -4x + -1x2 = 0 + 4x + -4x + x2 + -1x2
4 + -4x + -1x2 = 4x + -4x + x2 + -1x2

Combine like terms: 4x + -4x = 0
4 + -4x + -1x2 = 0 + x2 + -1x2
4 + -4x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
4 + -4x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-4 + 4x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 4x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 4x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 4x + x2 = 0 + 4
4x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
4x + x2 = 4

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 4 + 4

Reorder the terms:
4 + 4x + x2 = 4 + 4

Combine like terms: 4 + 4 = 8
4 + 4x + x2 = 8

Factor a perfect square on the left side:
(x + 2)(x + 2) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + 2) equal to 2.828427125 and -2.828427125.


Subproblem 1
x + 2 = 2.828427125

Simplifying
x + 2 = 2.828427125

Reorder the terms:
2 + x = 2.828427125

Solving
2 + x = 2.828427125

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = 2.828427125 + -2

Combine like terms: 2 + -2 = 0
0 + x = 2.828427125 + -2
x = 2.828427125 + -2

Combine like terms: 2.828427125 + -2 = 0.828427125
x = 0.828427125

Simplifying
x = 0.828427125


Subproblem 2
x + 2 = -2.828427125

Simplifying
x + 2 = -2.828427125

Reorder the terms:
2 + x = -2.828427125

Solving
2 + x = -2.828427125

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = -2.828427125 + -2

Combine like terms: 2 + -2 = 0
0 + x = -2.828427125 + -2
x = -2.828427125 + -2

Combine like terms: -2.828427125 + -2 = -4.828427125
x = -4.828427125

Simplifying
x = -4.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.828427125, -4.828427125}

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