4n+8=n^2+9

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Solution for 4n+8=n^2+9 equation:


Simplifying
4n + 8 = n2 + 9

Reorder the terms:
8 + 4n = n2 + 9

Reorder the terms:
8 + 4n = 9 + n2

Solving
8 + 4n = 9 + n2

Solving for variable 'n'.

Reorder the terms:
8 + -9 + 4n + -1n2 = 9 + n2 + -9 + -1n2

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

Reorder the terms:
-1 + 4n + -1n2 = 9 + -9 + n2 + -1n2

Combine like terms: 9 + -9 = 0
-1 + 4n + -1n2 = 0 + n2 + -1n2
-1 + 4n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-1 + 4n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -4n + n2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -4n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4n + n2 = 0 + -1
-4n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4n + n2 = -1

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = -1 + 4

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

Combine like terms: -1 + 4 = 3
4 + -4n + n2 = 3

Factor a perfect square on the left side:
(n + -2)(n + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + -2 = 1.732050808 Simplifying n + -2 = 1.732050808 Reorder the terms: -2 + n = 1.732050808 Solving -2 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + n = 1.732050808 + 2 n = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 n = 3.732050808 Simplifying n = 3.732050808

Subproblem 2

n + -2 = -1.732050808 Simplifying n + -2 = -1.732050808 Reorder the terms: -2 + n = -1.732050808 Solving -2 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + n = -1.732050808 + 2 n = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 n = 0.267949192 Simplifying n = 0.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.732050808, 0.267949192}

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