2x^2+10x=-1

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Solution for 2x^2+10x=-1 equation: 


Simplifying
2x2 + 10x = -1

Reorder the terms:
10x + 2x2 = -1

Solving
10x + 2x2 = -1

Solving for variable 'x'.

Reorder the terms:
1 + 10x + 2x2 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + 10x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 5x + x2 = 0 + -0.5
5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
5x + x2 = -0.5

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 = -0.5 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -0.5 + 6.25

Combine like terms: -0.5 + 6.25 = 5.75
6.25 + 5x + x2 = 5.75

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.397915762 and -2.397915762.


Subproblem 1
x + 2.5 = 2.397915762

Simplifying
x + 2.5 = 2.397915762

Reorder the terms:
2.5 + x = 2.397915762

Solving
2.5 + x = 2.397915762

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 = 2.397915762 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 2.397915762 + -2.5
x = 2.397915762 + -2.5

Combine like terms: 2.397915762 + -2.5 = -0.102084238
x = -0.102084238

Simplifying
x = -0.102084238


Subproblem 2
x + 2.5 = -2.397915762

Simplifying
x + 2.5 = -2.397915762

Reorder the terms:
2.5 + x = -2.397915762

Solving
2.5 + x = -2.397915762

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 = -2.397915762 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -2.397915762 + -2.5
x = -2.397915762 + -2.5

Combine like terms: -2.397915762 + -2.5 = -4.897915762
x = -4.897915762

Simplifying
x = -4.897915762


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.102084238, -4.897915762}

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