n(n-1)=4n+18

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Solution for n(n-1)=4n+18 equation:


Simplifying
n(n + -1) = 4n + 18

Reorder the terms:
n(-1 + n) = 4n + 18
(-1 * n + n * n) = 4n + 18
(-1n + n2) = 4n + 18

Reorder the terms:
-1n + n2 = 18 + 4n

Solving
-1n + n2 = 18 + 4n

Solving for variable 'n'.

Reorder the terms:
-18 + -1n + -4n + n2 = 18 + 4n + -18 + -4n

Combine like terms: -1n + -4n = -5n
-18 + -5n + n2 = 18 + 4n + -18 + -4n

Reorder the terms:
-18 + -5n + n2 = 18 + -18 + 4n + -4n

Combine like terms: 18 + -18 = 0
-18 + -5n + n2 = 0 + 4n + -4n
-18 + -5n + n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
-18 + -5n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + -5n + 18 + n2 = 0 + 18

Reorder the terms:
-18 + 18 + -5n + n2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + -5n + n2 = 0 + 18
-5n + n2 = 0 + 18

Combine like terms: 0 + 18 = 18
-5n + n2 = 18

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = 18 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 18 + 6.25

Combine like terms: 18 + 6.25 = 24.25
6.25 + -5n + n2 = 24.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(n + -2.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

n + -2.5 = 4.924428901 Simplifying n + -2.5 = 4.924428901 Reorder the terms: -2.5 + n = 4.924428901 Solving -2.5 + n = 4.924428901 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 4.924428901 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 4.924428901 + 2.5 n = 4.924428901 + 2.5 Combine like terms: 4.924428901 + 2.5 = 7.424428901 n = 7.424428901 Simplifying n = 7.424428901

Subproblem 2

n + -2.5 = -4.924428901 Simplifying n + -2.5 = -4.924428901 Reorder the terms: -2.5 + n = -4.924428901 Solving -2.5 + n = -4.924428901 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -4.924428901 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -4.924428901 + 2.5 n = -4.924428901 + 2.5 Combine like terms: -4.924428901 + 2.5 = -2.424428901 n = -2.424428901 Simplifying n = -2.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.424428901, -2.424428901}

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