2n^2=12n-1

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Solution for 2n^2=12n-1 equation:


Simplifying
2n2 = 12n + -1

Reorder the terms:
2n2 = -1 + 12n

Solving
2n2 = -1 + 12n

Solving for variable 'n'.

Reorder the terms:
1 + -12n + 2n2 = -1 + 12n + 1 + -12n

Reorder the terms:
1 + -12n + 2n2 = -1 + 1 + 12n + -12n

Combine like terms: -1 + 1 = 0
1 + -12n + 2n2 = 0 + 12n + -12n
1 + -12n + 2n2 = 12n + -12n

Combine like terms: 12n + -12n = 0
1 + -12n + 2n2 = 0

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

Divide each side by '2'.
0.5 + -6n + n2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -6n + -0.5 + n2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -6n + n2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -6n + n2 = 0 + -0.5
-6n + n2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-6n + n2 = -0.5

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = -0.5 + 9

Reorder the terms:
9 + -6n + n2 = -0.5 + 9

Combine like terms: -0.5 + 9 = 8.5
9 + -6n + n2 = 8.5

Factor a perfect square on the left side:
(n + -3)(n + -3) = 8.5

Calculate the square root of the right side: 2.915475947

Break this problem into two subproblems by setting 
(n + -3) equal to 2.915475947 and -2.915475947.

Subproblem 1

n + -3 = 2.915475947 Simplifying n + -3 = 2.915475947 Reorder the terms: -3 + n = 2.915475947 Solving -3 + n = 2.915475947 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.915475947 + 3 n = 2.915475947 + 3 Combine like terms: 2.915475947 + 3 = 5.915475947 n = 5.915475947 Simplifying n = 5.915475947

Subproblem 2

n + -3 = -2.915475947 Simplifying n + -3 = -2.915475947 Reorder the terms: -3 + n = -2.915475947 Solving -3 + n = -2.915475947 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.915475947 + 3 n = -2.915475947 + 3 Combine like terms: -2.915475947 + 3 = 0.084524053 n = 0.084524053 Simplifying n = 0.084524053

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.915475947, 0.084524053}

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