0=x^2+5x+1

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Solution for 0=x^2+5x+1 equation: 


Simplifying
0 = x2 + 5x + 1

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

Solving
0 = 1 + 5x + x2

Solving for variable 'x'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 5x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 5x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 5x + x2 = 0

Solving
1 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 5x + -1 + x2 = 0 + -1

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

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

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

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -1 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + 5x + x2 = 5.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
x + 2.5 = 2.291287847

Simplifying
x + 2.5 = 2.291287847

Reorder the terms:
2.5 + x = 2.291287847

Solving
2.5 + x = 2.291287847

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 2.291287847 + -2.5
x = 2.291287847 + -2.5

Combine like terms: 2.291287847 + -2.5 = -0.208712153
x = -0.208712153

Simplifying
x = -0.208712153


Subproblem 2
x + 2.5 = -2.291287847

Simplifying
x + 2.5 = -2.291287847

Reorder the terms:
2.5 + x = -2.291287847

Solving
2.5 + x = -2.291287847

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -2.291287847 + -2.5
x = -2.291287847 + -2.5

Combine like terms: -2.291287847 + -2.5 = -4.791287847
x = -4.791287847

Simplifying
x = -4.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.208712153, -4.791287847}
Solution
x = {-0.208712153, -4.791287847}

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