n^2+1=5n+5

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Solution for n^2+1=5n+5 equation:


Simplifying
n2 + 1 = 5n + 5

Reorder the terms:
1 + n2 = 5n + 5

Reorder the terms:
1 + n2 = 5 + 5n

Solving
1 + n2 = 5 + 5n

Solving for variable 'n'.

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

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -5n + 4 + n2 = 0 + 4

Reorder the terms:
-4 + 4 + -5n + n2 = 0 + 4

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

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

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 = 4 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 4 + 6.25

Combine like terms: 4 + 6.25 = 10.25
6.25 + -5n + n2 = 10.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + -2.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

n + -2.5 = 3.201562119 Simplifying n + -2.5 = 3.201562119 Reorder the terms: -2.5 + n = 3.201562119 Solving -2.5 + n = 3.201562119 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 = 3.201562119 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 3.201562119 + 2.5 n = 3.201562119 + 2.5 Combine like terms: 3.201562119 + 2.5 = 5.701562119 n = 5.701562119 Simplifying n = 5.701562119

Subproblem 2

n + -2.5 = -3.201562119 Simplifying n + -2.5 = -3.201562119 Reorder the terms: -2.5 + n = -3.201562119 Solving -2.5 + n = -3.201562119 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 = -3.201562119 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -3.201562119 + 2.5 n = -3.201562119 + 2.5 Combine like terms: -3.201562119 + 2.5 = -0.701562119 n = -0.701562119 Simplifying n = -0.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.701562119, -0.701562119}

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