2n^2=21+7n

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Solution for 2n^2=21+7n equation:


Simplifying
2n2 = 21 + 7n

Solving
2n2 = 21 + 7n

Solving for variable 'n'.

Reorder the terms:
-21 + -7n + 2n2 = 21 + 7n + -21 + -7n

Reorder the terms:
-21 + -7n + 2n2 = 21 + -21 + 7n + -7n

Combine like terms: 21 + -21 = 0
-21 + -7n + 2n2 = 0 + 7n + -7n
-21 + -7n + 2n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
-21 + -7n + 2n2 = 0

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

Divide each side by '2'.
-10.5 + -3.5n + n2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -3.5n + 10.5 + n2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -3.5n + n2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -3.5n + n2 = 0 + 10.5
-3.5n + n2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-3.5n + n2 = 10.5

The n term is -3.5n.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5n + 3.0625 + n2 = 10.5 + 3.0625

Reorder the terms:
3.0625 + -3.5n + n2 = 10.5 + 3.0625

Combine like terms: 10.5 + 3.0625 = 13.5625
3.0625 + -3.5n + n2 = 13.5625

Factor a perfect square on the left side:
(n + -1.75)(n + -1.75) = 13.5625

Calculate the square root of the right side: 3.682729966

Break this problem into two subproblems by setting 
(n + -1.75) equal to 3.682729966 and -3.682729966.

Subproblem 1

n + -1.75 = 3.682729966 Simplifying n + -1.75 = 3.682729966 Reorder the terms: -1.75 + n = 3.682729966 Solving -1.75 + n = 3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = 3.682729966 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 3.682729966 + 1.75 n = 3.682729966 + 1.75 Combine like terms: 3.682729966 + 1.75 = 5.432729966 n = 5.432729966 Simplifying n = 5.432729966

Subproblem 2

n + -1.75 = -3.682729966 Simplifying n + -1.75 = -3.682729966 Reorder the terms: -1.75 + n = -3.682729966 Solving -1.75 + n = -3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = -3.682729966 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -3.682729966 + 1.75 n = -3.682729966 + 1.75 Combine like terms: -3.682729966 + 1.75 = -1.932729966 n = -1.932729966 Simplifying n = -1.932729966

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.432729966, -1.932729966}

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