21n^2+8-4=6n+11

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Solution for 21n^2+8-4=6n+11 equation:


Simplifying
21n2 + 8 + -4 = 6n + 11

Reorder the terms:
8 + -4 + 21n2 = 6n + 11

Combine like terms: 8 + -4 = 4
4 + 21n2 = 6n + 11

Reorder the terms:
4 + 21n2 = 11 + 6n

Solving
4 + 21n2 = 11 + 6n

Solving for variable 'n'.

Reorder the terms:
4 + -11 + -6n + 21n2 = 11 + 6n + -11 + -6n

Combine like terms: 4 + -11 = -7
-7 + -6n + 21n2 = 11 + 6n + -11 + -6n

Reorder the terms:
-7 + -6n + 21n2 = 11 + -11 + 6n + -6n

Combine like terms: 11 + -11 = 0
-7 + -6n + 21n2 = 0 + 6n + -6n
-7 + -6n + 21n2 = 6n + -6n

Combine like terms: 6n + -6n = 0
-7 + -6n + 21n2 = 0

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

Divide each side by '21'.
-0.3333333333 + -0.2857142857n + n2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -0.2857142857n + 0.3333333333 + n2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -0.2857142857n + n2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -0.2857142857n + n2 = 0 + 0.3333333333
-0.2857142857n + n2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-0.2857142857n + n2 = 0.3333333333

The n term is -0.2857142857n.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857n + 0.02040816328 + n2 = 0.3333333333 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857n + n2 = 0.3333333333 + 0.02040816328

Combine like terms: 0.3333333333 + 0.02040816328 = 0.35374149658
0.02040816328 + -0.2857142857n + n2 = 0.35374149658

Factor a perfect square on the left side:
(n + -0.1428571429)(n + -0.1428571429) = 0.35374149658

Calculate the square root of the right side: 0.594761714

Break this problem into two subproblems by setting 
(n + -0.1428571429) equal to 0.594761714 and -0.594761714.

Subproblem 1

n + -0.1428571429 = 0.594761714 Simplifying n + -0.1428571429 = 0.594761714 Reorder the terms: -0.1428571429 + n = 0.594761714 Solving -0.1428571429 + n = 0.594761714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = 0.594761714 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = 0.594761714 + 0.1428571429 n = 0.594761714 + 0.1428571429 Combine like terms: 0.594761714 + 0.1428571429 = 0.7376188569 n = 0.7376188569 Simplifying n = 0.7376188569

Subproblem 2

n + -0.1428571429 = -0.594761714 Simplifying n + -0.1428571429 = -0.594761714 Reorder the terms: -0.1428571429 + n = -0.594761714 Solving -0.1428571429 + n = -0.594761714 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + n = -0.594761714 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + n = -0.594761714 + 0.1428571429 n = -0.594761714 + 0.1428571429 Combine like terms: -0.594761714 + 0.1428571429 = -0.4519045711 n = -0.4519045711 Simplifying n = -0.4519045711

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.7376188569, -0.4519045711}

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