10n^2-18=-9n

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Solution for 10n^2-18=-9n equation:


Simplifying
10n2 + -18 = -9n

Reorder the terms:
-18 + 10n2 = -9n

Solving
-18 + 10n2 = -9n

Solving for variable 'n'.

Reorder the terms:
-18 + 9n + 10n2 = -9n + 9n

Combine like terms: -9n + 9n = 0
-18 + 9n + 10n2 = 0

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

Divide each side by '10'.
-1.8 + 0.9n + n2 = 0

Move the constant term to the right:

Add '1.8' to each side of the equation.
-1.8 + 0.9n + 1.8 + n2 = 0 + 1.8

Reorder the terms:
-1.8 + 1.8 + 0.9n + n2 = 0 + 1.8

Combine like terms: -1.8 + 1.8 = 0.0
0.0 + 0.9n + n2 = 0 + 1.8
0.9n + n2 = 0 + 1.8

Combine like terms: 0 + 1.8 = 1.8
0.9n + n2 = 1.8

The n term is 0.9n.  Take half its coefficient (0.45).
Square it (0.2025) and add it to both sides.

Add '0.2025' to each side of the equation.
0.9n + 0.2025 + n2 = 1.8 + 0.2025

Reorder the terms:
0.2025 + 0.9n + n2 = 1.8 + 0.2025

Combine like terms: 1.8 + 0.2025 = 2.0025
0.2025 + 0.9n + n2 = 2.0025

Factor a perfect square on the left side:
(n + 0.45)(n + 0.45) = 2.0025

Calculate the square root of the right side: 1.41509717

Break this problem into two subproblems by setting 
(n + 0.45) equal to 1.41509717 and -1.41509717.

Subproblem 1

n + 0.45 = 1.41509717 Simplifying n + 0.45 = 1.41509717 Reorder the terms: 0.45 + n = 1.41509717 Solving 0.45 + n = 1.41509717 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + n = 1.41509717 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = 1.41509717 + -0.45 n = 1.41509717 + -0.45 Combine like terms: 1.41509717 + -0.45 = 0.96509717 n = 0.96509717 Simplifying n = 0.96509717

Subproblem 2

n + 0.45 = -1.41509717 Simplifying n + 0.45 = -1.41509717 Reorder the terms: 0.45 + n = -1.41509717 Solving 0.45 + n = -1.41509717 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + n = -1.41509717 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = -1.41509717 + -0.45 n = -1.41509717 + -0.45 Combine like terms: -1.41509717 + -0.45 = -1.86509717 n = -1.86509717 Simplifying n = -1.86509717

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.96509717, -1.86509717}

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