9n^2+10n=11

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Solution for 9n^2+10n=11 equation:


Simplifying
9n2 + 10n = 11

Reorder the terms:
10n + 9n2 = 11

Solving
10n + 9n2 = 11

Solving for variable 'n'.

Reorder the terms:
-11 + 10n + 9n2 = 11 + -11

Combine like terms: 11 + -11 = 0
-11 + 10n + 9n2 = 0

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

Divide each side by '9'.
-1.222222222 + 1.111111111n + n2 = 0

Move the constant term to the right:

Add '1.222222222' to each side of the equation.
-1.222222222 + 1.111111111n + 1.222222222 + n2 = 0 + 1.222222222

Reorder the terms:
-1.222222222 + 1.222222222 + 1.111111111n + n2 = 0 + 1.222222222

Combine like terms: -1.222222222 + 1.222222222 = 0.000000000
0.000000000 + 1.111111111n + n2 = 0 + 1.222222222
1.111111111n + n2 = 0 + 1.222222222

Combine like terms: 0 + 1.222222222 = 1.222222222
1.111111111n + n2 = 1.222222222

The n term is 1.111111111n.  Take half its coefficient (0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
1.111111111n + 0.3086419752 + n2 = 1.222222222 + 0.3086419752

Reorder the terms:
0.3086419752 + 1.111111111n + n2 = 1.222222222 + 0.3086419752

Combine like terms: 1.222222222 + 0.3086419752 = 1.5308641972
0.3086419752 + 1.111111111n + n2 = 1.5308641972

Factor a perfect square on the left side:
(n + 0.5555555555)(n + 0.5555555555) = 1.5308641972

Calculate the square root of the right side: 1.237280969

Break this problem into two subproblems by setting 
(n + 0.5555555555) equal to 1.237280969 and -1.237280969.

Subproblem 1

n + 0.5555555555 = 1.237280969 Simplifying n + 0.5555555555 = 1.237280969 Reorder the terms: 0.5555555555 + n = 1.237280969 Solving 0.5555555555 + n = 1.237280969 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + n = 1.237280969 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = 1.237280969 + -0.5555555555 n = 1.237280969 + -0.5555555555 Combine like terms: 1.237280969 + -0.5555555555 = 0.6817254135 n = 0.6817254135 Simplifying n = 0.6817254135

Subproblem 2

n + 0.5555555555 = -1.237280969 Simplifying n + 0.5555555555 = -1.237280969 Reorder the terms: 0.5555555555 + n = -1.237280969 Solving 0.5555555555 + n = -1.237280969 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + n = -1.237280969 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = -1.237280969 + -0.5555555555 n = -1.237280969 + -0.5555555555 Combine like terms: -1.237280969 + -0.5555555555 = -1.7928365245 n = -1.7928365245 Simplifying n = -1.7928365245

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.6817254135, -1.7928365245}

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