4x^2+2x+1=25

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Solution for 4x^2+2x+1=25 equation: 


Simplifying
4x2 + 2x + 1 = 25

Reorder the terms:
1 + 2x + 4x2 = 25

Solving
1 + 2x + 4x2 = 25

Solving for variable 'x'.

Reorder the terms:
1 + -25 + 2x + 4x2 = 25 + -25

Combine like terms: 1 + -25 = -24
-24 + 2x + 4x2 = 25 + -25

Combine like terms: 25 + -25 = 0
-24 + 2x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-12 + x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-12 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-12 + x + 2x2 = 0

Solving
-12 + x + 2x2 = 0

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

Divide each side by '2'.
-6 + 0.5x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 0.5x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 0.5x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 0.5x + x2 = 0 + 6
0.5x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
0.5x + x2 = 6

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 6 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 6 + 0.25

Combine like terms: 6 + 0.25 = 6.25
0.25 + 0.5x + x2 = 6.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 6.25

Calculate the square root of the right side: 2.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.5 and -2.5.


Subproblem 1
x + 0.5 = 2.5

Simplifying
x + 0.5 = 2.5

Reorder the terms:
0.5 + x = 2.5

Solving
0.5 + x = 2.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.5 + -0.5
x = 2.5 + -0.5

Combine like terms: 2.5 + -0.5 = 2
x = 2

Simplifying
x = 2


Subproblem 2
x + 0.5 = -2.5

Simplifying
x + 0.5 = -2.5

Reorder the terms:
0.5 + x = -2.5

Solving
0.5 + x = -2.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.5 + -0.5
x = -2.5 + -0.5

Combine like terms: -2.5 + -0.5 = -3
x = -3

Simplifying
x = -3


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2, -3}
Solution
x = {2, -3}

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