11n^2+6n-21=0

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Solution for 11n^2+6n-21=0 equation:


Simplifying
11n2 + 6n + -21 = 0

Reorder the terms:
-21 + 6n + 11n2 = 0

Solving
-21 + 6n + 11n2 = 0

Solving for variable 'n'.

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

Divide each side by '11'.
-1.909090909 + 0.5454545455n + n2 = 0

Move the constant term to the right:

Add '1.909090909' to each side of the equation.
-1.909090909 + 0.5454545455n + 1.909090909 + n2 = 0 + 1.909090909

Reorder the terms:
-1.909090909 + 1.909090909 + 0.5454545455n + n2 = 0 + 1.909090909

Combine like terms: -1.909090909 + 1.909090909 = 0.000000000
0.000000000 + 0.5454545455n + n2 = 0 + 1.909090909
0.5454545455n + n2 = 0 + 1.909090909

Combine like terms: 0 + 1.909090909 = 1.909090909
0.5454545455n + n2 = 1.909090909

The n term is 0.5454545455n.  Take half its coefficient (0.2727272728).
Square it (0.07438016533) and add it to both sides.

Add '0.07438016533' to each side of the equation.
0.5454545455n + 0.07438016533 + n2 = 1.909090909 + 0.07438016533

Reorder the terms:
0.07438016533 + 0.5454545455n + n2 = 1.909090909 + 0.07438016533

Combine like terms: 1.909090909 + 0.07438016533 = 1.98347107433
0.07438016533 + 0.5454545455n + n2 = 1.98347107433

Factor a perfect square on the left side:
(n + 0.2727272728)(n + 0.2727272728) = 1.98347107433

Calculate the square root of the right side: 1.40835758

Break this problem into two subproblems by setting 
(n + 0.2727272728) equal to 1.40835758 and -1.40835758.

Subproblem 1

n + 0.2727272728 = 1.40835758 Simplifying n + 0.2727272728 = 1.40835758 Reorder the terms: 0.2727272728 + n = 1.40835758 Solving 0.2727272728 + n = 1.40835758 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = 1.40835758 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = 1.40835758 + -0.2727272728 n = 1.40835758 + -0.2727272728 Combine like terms: 1.40835758 + -0.2727272728 = 1.1356303072 n = 1.1356303072 Simplifying n = 1.1356303072

Subproblem 2

n + 0.2727272728 = -1.40835758 Simplifying n + 0.2727272728 = -1.40835758 Reorder the terms: 0.2727272728 + n = -1.40835758 Solving 0.2727272728 + n = -1.40835758 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = -1.40835758 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = -1.40835758 + -0.2727272728 n = -1.40835758 + -0.2727272728 Combine like terms: -1.40835758 + -0.2727272728 = -1.6810848528 n = -1.6810848528 Simplifying n = -1.6810848528

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.1356303072, -1.6810848528}

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