4x^2+20x+10=0

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Solution for 4x^2+20x+10=0 equation: 


Simplifying
4x2 + 20x + 10 = 0

Reorder the terms:
10 + 20x + 4x2 = 0

Solving
10 + 20x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 10x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + 10x + 2x2)' equal to zero and attempt to solve:

Simplifying
5 + 10x + 2x2 = 0

Solving
5 + 10x + 2x2 = 0

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

Divide each side by '2'.
2.5 + 5x + x2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 5x + -2.5 + x2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 5x + x2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 5x + x2 = 0 + -2.5
5x + x2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
5x + x2 = -2.5

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 = -2.5 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -2.5 + 6.25

Combine like terms: -2.5 + 6.25 = 3.75
6.25 + 5x + x2 = 3.75

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.936491673 and -1.936491673.


Subproblem 1
x + 2.5 = 1.936491673

Simplifying
x + 2.5 = 1.936491673

Reorder the terms:
2.5 + x = 1.936491673

Solving
2.5 + x = 1.936491673

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 = 1.936491673 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 1.936491673 + -2.5
x = 1.936491673 + -2.5

Combine like terms: 1.936491673 + -2.5 = -0.563508327
x = -0.563508327

Simplifying
x = -0.563508327


Subproblem 2
x + 2.5 = -1.936491673

Simplifying
x + 2.5 = -1.936491673

Reorder the terms:
2.5 + x = -1.936491673

Solving
2.5 + x = -1.936491673

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 = -1.936491673 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -1.936491673 + -2.5
x = -1.936491673 + -2.5

Combine like terms: -1.936491673 + -2.5 = -4.436491673
x = -4.436491673

Simplifying
x = -4.436491673


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.563508327, -4.436491673}
Solution
x = {-0.563508327, -4.436491673}

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