3n^2=15+n

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Solution for 3n^2=15+n equation:


Simplifying
3n2 = 15 + n

Solving
3n2 = 15 + n

Solving for variable 'n'.

Reorder the terms:
-15 + -1n + 3n2 = 15 + n + -15 + -1n

Reorder the terms:
-15 + -1n + 3n2 = 15 + -15 + n + -1n

Combine like terms: 15 + -15 = 0
-15 + -1n + 3n2 = 0 + n + -1n
-15 + -1n + 3n2 = n + -1n

Combine like terms: n + -1n = 0
-15 + -1n + 3n2 = 0

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

Divide each side by '3'.
-5 + -0.3333333333n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.3333333333n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.3333333333n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.3333333333n + n2 = 0 + 5
-0.3333333333n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.3333333333n + n2 = 5

The n term is -0.3333333333n.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333n + 0.02777777779 + n2 = 5 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333n + n2 = 5 + 0.02777777779

Combine like terms: 5 + 0.02777777779 = 5.02777777779
0.02777777779 + -0.3333333333n + n2 = 5.02777777779

Factor a perfect square on the left side:
(n + -0.1666666667)(n + -0.1666666667) = 5.02777777779

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(n + -0.1666666667) equal to 2.242270675 and -2.242270675.

Subproblem 1

n + -0.1666666667 = 2.242270675 Simplifying n + -0.1666666667 = 2.242270675 Reorder the terms: -0.1666666667 + n = 2.242270675 Solving -0.1666666667 + n = 2.242270675 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = 2.242270675 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = 2.242270675 + 0.1666666667 n = 2.242270675 + 0.1666666667 Combine like terms: 2.242270675 + 0.1666666667 = 2.4089373417 n = 2.4089373417 Simplifying n = 2.4089373417

Subproblem 2

n + -0.1666666667 = -2.242270675 Simplifying n + -0.1666666667 = -2.242270675 Reorder the terms: -0.1666666667 + n = -2.242270675 Solving -0.1666666667 + n = -2.242270675 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = -2.242270675 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = -2.242270675 + 0.1666666667 n = -2.242270675 + 0.1666666667 Combine like terms: -2.242270675 + 0.1666666667 = -2.0756040083 n = -2.0756040083 Simplifying n = -2.0756040083

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.4089373417, -2.0756040083}

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