4n^2=4n+1

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Solution for 4n^2=4n+1 equation:


Simplifying
4n2 = 4n + 1

Reorder the terms:
4n2 = 1 + 4n

Solving
4n2 = 1 + 4n

Solving for variable 'n'.

Reorder the terms:
-1 + -4n + 4n2 = 1 + 4n + -1 + -4n

Reorder the terms:
-1 + -4n + 4n2 = 1 + -1 + 4n + -4n

Combine like terms: 1 + -1 = 0
-1 + -4n + 4n2 = 0 + 4n + -4n
-1 + -4n + 4n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
-1 + -4n + 4n2 = 0

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

Divide each side by '4'.
-0.25 + -1n + n2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1n + 0.25 + n2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1n + n2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1n + n2 = 0 + 0.25
-1n + n2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1n + n2 = 0.25

The n term is -1n.  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.
-1n + 0.25 + n2 = 0.25 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 0.25 + 0.25

Combine like terms: 0.25 + 0.25 = 0.5
0.25 + -1n + n2 = 0.5

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(n + -0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

n + -0.5 = 0.707106781 Simplifying n + -0.5 = 0.707106781 Reorder the terms: -0.5 + n = 0.707106781 Solving -0.5 + n = 0.707106781 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 0.707106781 + 0.5 n = 0.707106781 + 0.5 Combine like terms: 0.707106781 + 0.5 = 1.207106781 n = 1.207106781 Simplifying n = 1.207106781

Subproblem 2

n + -0.5 = -0.707106781 Simplifying n + -0.5 = -0.707106781 Reorder the terms: -0.5 + n = -0.707106781 Solving -0.5 + n = -0.707106781 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -0.707106781 + 0.5 n = -0.707106781 + 0.5 Combine like terms: -0.707106781 + 0.5 = -0.207106781 n = -0.207106781 Simplifying n = -0.207106781

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.207106781, -0.207106781}

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