n^2=18-9n

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


Simplifying
n2 = 18 + -9n

Solving
n2 = 18 + -9n

Solving for variable 'n'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 9n + 18 + n2 = 0 + 18

Reorder the terms:
-18 + 18 + 9n + n2 = 0 + 18

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

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

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = 18 + 20.25

Reorder the terms:
20.25 + 9n + n2 = 18 + 20.25

Combine like terms: 18 + 20.25 = 38.25
20.25 + 9n + n2 = 38.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 38.25

Calculate the square root of the right side: 6.184658438

Break this problem into two subproblems by setting 
(n + 4.5) equal to 6.184658438 and -6.184658438.

Subproblem 1

n + 4.5 = 6.184658438 Simplifying n + 4.5 = 6.184658438 Reorder the terms: 4.5 + n = 6.184658438 Solving 4.5 + n = 6.184658438 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 6.184658438 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 6.184658438 + -4.5 n = 6.184658438 + -4.5 Combine like terms: 6.184658438 + -4.5 = 1.684658438 n = 1.684658438 Simplifying n = 1.684658438

Subproblem 2

n + 4.5 = -6.184658438 Simplifying n + 4.5 = -6.184658438 Reorder the terms: 4.5 + n = -6.184658438 Solving 4.5 + n = -6.184658438 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -6.184658438 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -6.184658438 + -4.5 n = -6.184658438 + -4.5 Combine like terms: -6.184658438 + -4.5 = -10.684658438 n = -10.684658438 Simplifying n = -10.684658438

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.684658438, -10.684658438}

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