n^2-(n+4)=8n

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Solution for n^2-(n+4)=8n equation:


Simplifying
n2 + -1(n + 4) = 8n

Reorder the terms:
n2 + -1(4 + n) = 8n
n2 + (4 * -1 + n * -1) = 8n
n2 + (-4 + -1n) = 8n

Reorder the terms:
-4 + -1n + n2 = 8n

Solving
-4 + -1n + n2 = 8n

Solving for variable 'n'.

Reorder the terms:
-4 + -1n + -8n + n2 = 8n + -8n

Combine like terms: -1n + -8n = -9n
-4 + -9n + n2 = 8n + -8n

Combine like terms: 8n + -8n = 0
-4 + -9n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -9n + 4 + n2 = 0 + 4

Reorder the terms:
-4 + 4 + -9n + n2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -9n + n2 = 0 + 4
-9n + n2 = 0 + 4

Combine like terms: 0 + 4 = 4
-9n + n2 = 4

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 = 4 + 20.25

Reorder the terms:
20.25 + -9n + n2 = 4 + 20.25

Combine like terms: 4 + 20.25 = 24.25
20.25 + -9n + n2 = 24.25

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(n + -4.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

n + -4.5 = 4.924428901 Simplifying n + -4.5 = 4.924428901 Reorder the terms: -4.5 + n = 4.924428901 Solving -4.5 + n = 4.924428901 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 = 4.924428901 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 4.924428901 + 4.5 n = 4.924428901 + 4.5 Combine like terms: 4.924428901 + 4.5 = 9.424428901 n = 9.424428901 Simplifying n = 9.424428901

Subproblem 2

n + -4.5 = -4.924428901 Simplifying n + -4.5 = -4.924428901 Reorder the terms: -4.5 + n = -4.924428901 Solving -4.5 + n = -4.924428901 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 = -4.924428901 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -4.924428901 + 4.5 n = -4.924428901 + 4.5 Combine like terms: -4.924428901 + 4.5 = -0.424428901 n = -0.424428901 Simplifying n = -0.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.424428901, -0.424428901}

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