n^2=4n-2

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Solution for n^2=4n-2 equation:


Simplifying
n2 = 4n + -2

Reorder the terms:
n2 = -2 + 4n

Solving
n2 = -2 + 4n

Solving for variable 'n'.

Reorder the terms:
2 + -4n + n2 = -2 + 4n + 2 + -4n

Reorder the terms:
2 + -4n + n2 = -2 + 2 + 4n + -4n

Combine like terms: -2 + 2 = 0
2 + -4n + n2 = 0 + 4n + -4n
2 + -4n + n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
2 + -4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4n + -2 + n2 = 0 + -2

Reorder the terms:
2 + -2 + -4n + n2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4n + n2 = 0 + -2
-4n + n2 = 0 + -2

Combine like terms: 0 + -2 = -2
-4n + n2 = -2

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = -2 + 4

Reorder the terms:
4 + -4n + n2 = -2 + 4

Combine like terms: -2 + 4 = 2
4 + -4n + n2 = 2

Factor a perfect square on the left side:
(n + -2)(n + -2) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(n + -2) equal to 1.414213562 and -1.414213562.

Subproblem 1

n + -2 = 1.414213562 Simplifying n + -2 = 1.414213562 Reorder the terms: -2 + n = 1.414213562 Solving -2 + n = 1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + n = 1.414213562 + 2 n = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 n = 3.414213562 Simplifying n = 3.414213562

Subproblem 2

n + -2 = -1.414213562 Simplifying n + -2 = -1.414213562 Reorder the terms: -2 + n = -1.414213562 Solving -2 + n = -1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + n = -1.414213562 + 2 n = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 n = 0.585786438 Simplifying n = 0.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.414213562, 0.585786438}

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