2n^2=n+2

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


Simplifying
2n2 = n + 2

Reorder the terms:
2n2 = 2 + n

Solving
2n2 = 2 + n

Solving for variable 'n'.

Reorder the terms:
-2 + -1n + 2n2 = 2 + n + -2 + -1n

Reorder the terms:
-2 + -1n + 2n2 = 2 + -2 + n + -1n

Combine like terms: 2 + -2 = 0
-2 + -1n + 2n2 = 0 + n + -1n
-2 + -1n + 2n2 = n + -1n

Combine like terms: n + -1n = 0
-2 + -1n + 2n2 = 0

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

Divide each side by '2'.
-1 + -0.5n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.5n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.5n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.5n + n2 = 0 + 1
-0.5n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.5n + n2 = 1

The n term is -0.5n.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5n + 0.0625 + n2 = 1 + 0.0625

Reorder the terms:
0.0625 + -0.5n + n2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + -0.5n + n2 = 1.0625

Factor a perfect square on the left side:
(n + -0.25)(n + -0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(n + -0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

n + -0.25 = 1.030776406 Simplifying n + -0.25 = 1.030776406 Reorder the terms: -0.25 + n = 1.030776406 Solving -0.25 + n = 1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 1.030776406 + 0.25 n = 1.030776406 + 0.25 Combine like terms: 1.030776406 + 0.25 = 1.280776406 n = 1.280776406 Simplifying n = 1.280776406

Subproblem 2

n + -0.25 = -1.030776406 Simplifying n + -0.25 = -1.030776406 Reorder the terms: -0.25 + n = -1.030776406 Solving -0.25 + n = -1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -1.030776406 + 0.25 n = -1.030776406 + 0.25 Combine like terms: -1.030776406 + 0.25 = -0.780776406 n = -0.780776406 Simplifying n = -0.780776406

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.280776406, -0.780776406}

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