5n+15=n^2+2

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Solution for 5n+15=n^2+2 equation:


Simplifying
5n + 15 = n2 + 2

Reorder the terms:
15 + 5n = n2 + 2

Reorder the terms:
15 + 5n = 2 + n2

Solving
15 + 5n = 2 + n2

Solving for variable 'n'.

Reorder the terms:
15 + -2 + 5n + -1n2 = 2 + n2 + -2 + -1n2

Combine like terms: 15 + -2 = 13
13 + 5n + -1n2 = 2 + n2 + -2 + -1n2

Reorder the terms:
13 + 5n + -1n2 = 2 + -2 + n2 + -1n2

Combine like terms: 2 + -2 = 0
13 + 5n + -1n2 = 0 + n2 + -1n2
13 + 5n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
13 + 5n + -1n2 = 0

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

Divide each side by '-1'.
-13 + -5n + n2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -5n + 13 + n2 = 0 + 13

Reorder the terms:
-13 + 13 + -5n + n2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -5n + n2 = 0 + 13
-5n + n2 = 0 + 13

Combine like terms: 0 + 13 = 13
-5n + n2 = 13

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 = 13 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 13 + 6.25

Combine like terms: 13 + 6.25 = 19.25
6.25 + -5n + n2 = 19.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(n + -2.5) equal to 4.387482194 and -4.387482194.

Subproblem 1

n + -2.5 = 4.387482194 Simplifying n + -2.5 = 4.387482194 Reorder the terms: -2.5 + n = 4.387482194 Solving -2.5 + n = 4.387482194 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 = 4.387482194 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 4.387482194 + 2.5 n = 4.387482194 + 2.5 Combine like terms: 4.387482194 + 2.5 = 6.887482194 n = 6.887482194 Simplifying n = 6.887482194

Subproblem 2

n + -2.5 = -4.387482194 Simplifying n + -2.5 = -4.387482194 Reorder the terms: -2.5 + n = -4.387482194 Solving -2.5 + n = -4.387482194 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 = -4.387482194 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -4.387482194 + 2.5 n = -4.387482194 + 2.5 Combine like terms: -4.387482194 + 2.5 = -1.887482194 n = -1.887482194 Simplifying n = -1.887482194

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.887482194, -1.887482194}

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