4x^2+10x+25=41

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Solution for 4x^2+10x+25=41 equation: 


Simplifying
4x2 + 10x + 25 = 41

Reorder the terms:
25 + 10x + 4x2 = 41

Solving
25 + 10x + 4x2 = 41

Solving for variable 'x'.

Reorder the terms:
25 + -41 + 10x + 4x2 = 41 + -41

Combine like terms: 25 + -41 = -16
-16 + 10x + 4x2 = 41 + -41

Combine like terms: 41 + -41 = 0
-16 + 10x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + 5x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-8 + 5x + 2x2)' equal to zero and attempt to solve:

Simplifying
-8 + 5x + 2x2 = 0

Solving
-8 + 5x + 2x2 = 0

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

Divide each side by '2'.
-4 + 2.5x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2.5x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 2.5x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2.5x + x2 = 0 + 4
2.5x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
2.5x + x2 = 4

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 4 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 4 + 1.5625

Combine like terms: 4 + 1.5625 = 5.5625
1.5625 + 2.5x + x2 = 5.5625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(x + 1.25) equal to 2.358495283 and -2.358495283.


Subproblem 1
x + 1.25 = 2.358495283

Simplifying
x + 1.25 = 2.358495283

Reorder the terms:
1.25 + x = 2.358495283

Solving
1.25 + x = 2.358495283

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = 2.358495283 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 2.358495283 + -1.25
x = 2.358495283 + -1.25

Combine like terms: 2.358495283 + -1.25 = 1.108495283
x = 1.108495283

Simplifying
x = 1.108495283


Subproblem 2
x + 1.25 = -2.358495283

Simplifying
x + 1.25 = -2.358495283

Reorder the terms:
1.25 + x = -2.358495283

Solving
1.25 + x = -2.358495283

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = -2.358495283 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -2.358495283 + -1.25
x = -2.358495283 + -1.25

Combine like terms: -2.358495283 + -1.25 = -3.608495283
x = -3.608495283

Simplifying
x = -3.608495283


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.108495283, -3.608495283}
Solution
x = {1.108495283, -3.608495283}

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