9n^2-10n=20

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Solution for 9n^2-10n=20 equation:


Simplifying
9n2 + -10n = 20

Reorder the terms:
-10n + 9n2 = 20

Solving
-10n + 9n2 = 20

Solving for variable 'n'.

Reorder the terms:
-20 + -10n + 9n2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + -10n + 9n2 = 0

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

Divide each side by '9'.
-2.222222222 + -1.111111111n + n2 = 0

Move the constant term to the right:

Add '2.222222222' to each side of the equation.
-2.222222222 + -1.111111111n + 2.222222222 + n2 = 0 + 2.222222222

Reorder the terms:
-2.222222222 + 2.222222222 + -1.111111111n + n2 = 0 + 2.222222222

Combine like terms: -2.222222222 + 2.222222222 = 0.000000000
0.000000000 + -1.111111111n + n2 = 0 + 2.222222222
-1.111111111n + n2 = 0 + 2.222222222

Combine like terms: 0 + 2.222222222 = 2.222222222
-1.111111111n + n2 = 2.222222222

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 = 2.222222222 + 0.3086419752

Reorder the terms:
0.3086419752 + -1.111111111n + n2 = 2.222222222 + 0.3086419752

Combine like terms: 2.222222222 + 0.3086419752 = 2.5308641972
0.3086419752 + -1.111111111n + n2 = 2.5308641972

Factor a perfect square on the left side:
(n + -0.5555555555)(n + -0.5555555555) = 2.5308641972

Calculate the square root of the right side: 1.590869007

Break this problem into two subproblems by setting 
(n + -0.5555555555) equal to 1.590869007 and -1.590869007.

Subproblem 1

n + -0.5555555555 = 1.590869007 Simplifying n + -0.5555555555 = 1.590869007 Reorder the terms: -0.5555555555 + n = 1.590869007 Solving -0.5555555555 + n = 1.590869007 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 = 1.590869007 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = 1.590869007 + 0.5555555555 n = 1.590869007 + 0.5555555555 Combine like terms: 1.590869007 + 0.5555555555 = 2.1464245625 n = 2.1464245625 Simplifying n = 2.1464245625

Subproblem 2

n + -0.5555555555 = -1.590869007 Simplifying n + -0.5555555555 = -1.590869007 Reorder the terms: -0.5555555555 + n = -1.590869007 Solving -0.5555555555 + n = -1.590869007 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 = -1.590869007 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = -1.590869007 + 0.5555555555 n = -1.590869007 + 0.5555555555 Combine like terms: -1.590869007 + 0.5555555555 = -1.0353134515 n = -1.0353134515 Simplifying n = -1.0353134515

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.1464245625, -1.0353134515}

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