10x^2+55x+5=0

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Solution for 10x^2+55x+5=0 equation: 


Simplifying
10x2 + 55x + 5 = 0

Reorder the terms:
5 + 55x + 10x2 = 0

Solving
5 + 55x + 10x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '5'.
5(1 + 11x + 2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(1 + 11x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + 11x + 2x2 = 0

Solving
1 + 11x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 5.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 5.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 5.5x + x2 = 0 + -0.5
5.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
5.5x + x2 = -0.5

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -0.5 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -0.5 + 7.5625

Combine like terms: -0.5 + 7.5625 = 7.0625
7.5625 + 5.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + 2.75) equal to 2.657536453 and -2.657536453.


Subproblem 1
x + 2.75 = 2.657536453

Simplifying
x + 2.75 = 2.657536453

Reorder the terms:
2.75 + x = 2.657536453

Solving
2.75 + x = 2.657536453

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 2.657536453 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 2.657536453 + -2.75
x = 2.657536453 + -2.75

Combine like terms: 2.657536453 + -2.75 = -0.092463547
x = -0.092463547

Simplifying
x = -0.092463547


Subproblem 2
x + 2.75 = -2.657536453

Simplifying
x + 2.75 = -2.657536453

Reorder the terms:
2.75 + x = -2.657536453

Solving
2.75 + x = -2.657536453

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -2.657536453 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -2.657536453 + -2.75
x = -2.657536453 + -2.75

Combine like terms: -2.657536453 + -2.75 = -5.407536453
x = -5.407536453

Simplifying
x = -5.407536453


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.092463547, -5.407536453}
Solution
x = {-0.092463547, -5.407536453}

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