9n^2-7n-21=0

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Solution for 9n^2-7n-21=0 equation:


Simplifying
9n2 + -7n + -21 = 0

Reorder the terms:
-21 + -7n + 9n2 = 0

Solving
-21 + -7n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
-2.333333333 + -0.7777777778n + n2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -0.7777777778n + 2.333333333 + n2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -0.7777777778n + n2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -0.7777777778n + n2 = 0 + 2.333333333
-0.7777777778n + n2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-0.7777777778n + n2 = 2.333333333

The n term is -0.7777777778n.  Take half its coefficient (-0.3888888889).
Square it (0.1512345679) and add it to both sides.

Add '0.1512345679' to each side of the equation.
-0.7777777778n + 0.1512345679 + n2 = 2.333333333 + 0.1512345679

Reorder the terms:
0.1512345679 + -0.7777777778n + n2 = 2.333333333 + 0.1512345679

Combine like terms: 2.333333333 + 0.1512345679 = 2.4845679009
0.1512345679 + -0.7777777778n + n2 = 2.4845679009

Factor a perfect square on the left side:
(n + -0.3888888889)(n + -0.3888888889) = 2.4845679009

Calculate the square root of the right side: 1.576251218

Break this problem into two subproblems by setting 
(n + -0.3888888889) equal to 1.576251218 and -1.576251218.

Subproblem 1

n + -0.3888888889 = 1.576251218 Simplifying n + -0.3888888889 = 1.576251218 Reorder the terms: -0.3888888889 + n = 1.576251218 Solving -0.3888888889 + n = 1.576251218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3888888889' to each side of the equation. -0.3888888889 + 0.3888888889 + n = 1.576251218 + 0.3888888889 Combine like terms: -0.3888888889 + 0.3888888889 = 0.0000000000 0.0000000000 + n = 1.576251218 + 0.3888888889 n = 1.576251218 + 0.3888888889 Combine like terms: 1.576251218 + 0.3888888889 = 1.9651401069 n = 1.9651401069 Simplifying n = 1.9651401069

Subproblem 2

n + -0.3888888889 = -1.576251218 Simplifying n + -0.3888888889 = -1.576251218 Reorder the terms: -0.3888888889 + n = -1.576251218 Solving -0.3888888889 + n = -1.576251218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3888888889' to each side of the equation. -0.3888888889 + 0.3888888889 + n = -1.576251218 + 0.3888888889 Combine like terms: -0.3888888889 + 0.3888888889 = 0.0000000000 0.0000000000 + n = -1.576251218 + 0.3888888889 n = -1.576251218 + 0.3888888889 Combine like terms: -1.576251218 + 0.3888888889 = -1.1873623291 n = -1.1873623291 Simplifying n = -1.1873623291

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.9651401069, -1.1873623291}

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