n^2-11=3-n

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Solution for n^2-11=3-n equation:


Simplifying
n2 + -11 = 3 + -1n

Reorder the terms:
-11 + n2 = 3 + -1n

Solving
-11 + n2 = 3 + -1n

Solving for variable 'n'.

Reorder the terms:
-11 + -3 + n + n2 = 3 + -1n + -3 + n

Combine like terms: -11 + -3 = -14
-14 + n + n2 = 3 + -1n + -3 + n

Reorder the terms:
-14 + n + n2 = 3 + -3 + -1n + n

Combine like terms: 3 + -3 = 0
-14 + n + n2 = 0 + -1n + n
-14 + n + n2 = -1n + n

Combine like terms: -1n + n = 0
-14 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + n + 14 + n2 = 0 + 14

Reorder the terms:
-14 + 14 + n + n2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + n + n2 = 0 + 14
n + n2 = 0 + 14

Combine like terms: 0 + 14 = 14
n + n2 = 14

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 14 + 0.25

Reorder the terms:
0.25 + n + n2 = 14 + 0.25

Combine like terms: 14 + 0.25 = 14.25
0.25 + n + n2 = 14.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + 0.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

n + 0.5 = 3.774917218 Simplifying n + 0.5 = 3.774917218 Reorder the terms: 0.5 + n = 3.774917218 Solving 0.5 + n = 3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.774917218 + -0.5 n = 3.774917218 + -0.5 Combine like terms: 3.774917218 + -0.5 = 3.274917218 n = 3.274917218 Simplifying n = 3.274917218

Subproblem 2

n + 0.5 = -3.774917218 Simplifying n + 0.5 = -3.774917218 Reorder the terms: 0.5 + n = -3.774917218 Solving 0.5 + n = -3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.774917218 + -0.5 n = -3.774917218 + -0.5 Combine like terms: -3.774917218 + -0.5 = -4.274917218 n = -4.274917218 Simplifying n = -4.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.274917218, -4.274917218}

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