n^2=n+18

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Solution for n^2=n+18 equation:


Simplifying
n2 = n + 18

Reorder the terms:
n2 = 18 + n

Solving
n2 = 18 + n

Solving for variable 'n'.

Reorder the terms:
-18 + -1n + n2 = 18 + n + -18 + -1n

Reorder the terms:
-18 + -1n + n2 = 18 + -18 + n + -1n

Combine like terms: 18 + -18 = 0
-18 + -1n + n2 = 0 + n + -1n
-18 + -1n + n2 = n + -1n

Combine like terms: n + -1n = 0
-18 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + -1n + 18 + n2 = 0 + 18

Reorder the terms:
-18 + 18 + -1n + n2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + -1n + n2 = 0 + 18
-1n + n2 = 0 + 18

Combine like terms: 0 + 18 = 18
-1n + n2 = 18

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 18 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 18 + 0.25

Combine like terms: 18 + 0.25 = 18.25
0.25 + -1n + n2 = 18.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + -0.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

n + -0.5 = 4.272001873 Simplifying n + -0.5 = 4.272001873 Reorder the terms: -0.5 + n = 4.272001873 Solving -0.5 + n = 4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 4.272001873 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 4.272001873 + 0.5 n = 4.272001873 + 0.5 Combine like terms: 4.272001873 + 0.5 = 4.772001873 n = 4.772001873 Simplifying n = 4.772001873

Subproblem 2

n + -0.5 = -4.272001873 Simplifying n + -0.5 = -4.272001873 Reorder the terms: -0.5 + n = -4.272001873 Solving -0.5 + n = -4.272001873 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -4.272001873 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -4.272001873 + 0.5 n = -4.272001873 + 0.5 Combine like terms: -4.272001873 + 0.5 = -3.772001873 n = -3.772001873 Simplifying n = -3.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.772001873, -3.772001873}

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