6n^2=21-2n

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Solution for 6n^2=21-2n equation:


Simplifying
6n2 = 21 + -2n

Solving
6n2 = 21 + -2n

Solving for variable 'n'.

Reorder the terms:
-21 + 2n + 6n2 = 21 + -2n + -21 + 2n

Reorder the terms:
-21 + 2n + 6n2 = 21 + -21 + -2n + 2n

Combine like terms: 21 + -21 = 0
-21 + 2n + 6n2 = 0 + -2n + 2n
-21 + 2n + 6n2 = -2n + 2n

Combine like terms: -2n + 2n = 0
-21 + 2n + 6n2 = 0

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

Divide each side by '6'.
-3.5 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 0.3333333333n + 3.5 + n2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 0.3333333333n + n2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 0.3333333333n + n2 = 0 + 3.5
0.3333333333n + n2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
0.3333333333n + n2 = 3.5

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 = 3.5 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333n + n2 = 3.5 + 0.02777777779

Combine like terms: 3.5 + 0.02777777779 = 3.52777777779
0.02777777779 + 0.3333333333n + n2 = 3.52777777779

Factor a perfect square on the left side:
(n + 0.1666666667)(n + 0.1666666667) = 3.52777777779

Calculate the square root of the right side: 1.878237945

Break this problem into two subproblems by setting 
(n + 0.1666666667) equal to 1.878237945 and -1.878237945.

Subproblem 1

n + 0.1666666667 = 1.878237945 Simplifying n + 0.1666666667 = 1.878237945 Reorder the terms: 0.1666666667 + n = 1.878237945 Solving 0.1666666667 + n = 1.878237945 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 = 1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + n = 1.878237945 + -0.1666666667 n = 1.878237945 + -0.1666666667 Combine like terms: 1.878237945 + -0.1666666667 = 1.7115712783 n = 1.7115712783 Simplifying n = 1.7115712783

Subproblem 2

n + 0.1666666667 = -1.878237945 Simplifying n + 0.1666666667 = -1.878237945 Reorder the terms: 0.1666666667 + n = -1.878237945 Solving 0.1666666667 + n = -1.878237945 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 = -1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + n = -1.878237945 + -0.1666666667 n = -1.878237945 + -0.1666666667 Combine like terms: -1.878237945 + -0.1666666667 = -2.0449046117 n = -2.0449046117 Simplifying n = -2.0449046117

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.7115712783, -2.0449046117}

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