11n^2-3n-21=0

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Solution for 11n^2-3n-21=0 equation:


Simplifying
11n2 + -3n + -21 = 0

Reorder the terms:
-21 + -3n + 11n2 = 0

Solving
-21 + -3n + 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 + -0.2727272727n + n2 = 0

Move the constant term to the right:

Add '1.909090909' to each side of the equation.
-1.909090909 + -0.2727272727n + 1.909090909 + n2 = 0 + 1.909090909

Reorder the terms:
-1.909090909 + 1.909090909 + -0.2727272727n + n2 = 0 + 1.909090909

Combine like terms: -1.909090909 + 1.909090909 = 0.000000000
0.000000000 + -0.2727272727n + n2 = 0 + 1.909090909
-0.2727272727n + n2 = 0 + 1.909090909

Combine like terms: 0 + 1.909090909 = 1.909090909
-0.2727272727n + n2 = 1.909090909

The n term is -0.2727272727n.  Take half its coefficient (-0.1363636364).
Square it (0.01859504133) and add it to both sides.

Add '0.01859504133' to each side of the equation.
-0.2727272727n + 0.01859504133 + n2 = 1.909090909 + 0.01859504133

Reorder the terms:
0.01859504133 + -0.2727272727n + n2 = 1.909090909 + 0.01859504133

Combine like terms: 1.909090909 + 0.01859504133 = 1.92768595033
0.01859504133 + -0.2727272727n + n2 = 1.92768595033

Factor a perfect square on the left side:
(n + -0.1363636364)(n + -0.1363636364) = 1.92768595033

Calculate the square root of the right side: 1.388411304

Break this problem into two subproblems by setting 
(n + -0.1363636364) equal to 1.388411304 and -1.388411304.

Subproblem 1

n + -0.1363636364 = 1.388411304 Simplifying n + -0.1363636364 = 1.388411304 Reorder the terms: -0.1363636364 + n = 1.388411304 Solving -0.1363636364 + n = 1.388411304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1363636364' to each side of the equation. -0.1363636364 + 0.1363636364 + n = 1.388411304 + 0.1363636364 Combine like terms: -0.1363636364 + 0.1363636364 = 0.0000000000 0.0000000000 + n = 1.388411304 + 0.1363636364 n = 1.388411304 + 0.1363636364 Combine like terms: 1.388411304 + 0.1363636364 = 1.5247749404 n = 1.5247749404 Simplifying n = 1.5247749404

Subproblem 2

n + -0.1363636364 = -1.388411304 Simplifying n + -0.1363636364 = -1.388411304 Reorder the terms: -0.1363636364 + n = -1.388411304 Solving -0.1363636364 + n = -1.388411304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1363636364' to each side of the equation. -0.1363636364 + 0.1363636364 + n = -1.388411304 + 0.1363636364 Combine like terms: -0.1363636364 + 0.1363636364 = 0.0000000000 0.0000000000 + n = -1.388411304 + 0.1363636364 n = -1.388411304 + 0.1363636364 Combine like terms: -1.388411304 + 0.1363636364 = -1.2520476676 n = -1.2520476676 Simplifying n = -1.2520476676

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.5247749404, -1.2520476676}

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