n^2-n=10

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Solution for n^2-n=10 equation:


Simplifying
n2 + -1n = 10

Reorder the terms:
-1n + n2 = 10

Solving
-1n + n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + -1n + n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -1n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + -1n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -1n + n2 = 0 + 10
-1n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
-1n + n2 = 10

The n term is -1n.  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.
-1n + 0.25 + n2 = 10 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 10 + 0.25

Combine like terms: 10 + 0.25 = 10.25
0.25 + -1n + n2 = 10.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(n + -0.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

n + -0.5 = 3.201562119 Simplifying n + -0.5 = 3.201562119 Reorder the terms: -0.5 + n = 3.201562119 Solving -0.5 + n = 3.201562119 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 = 3.201562119 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 3.201562119 + 0.5 n = 3.201562119 + 0.5 Combine like terms: 3.201562119 + 0.5 = 3.701562119 n = 3.701562119 Simplifying n = 3.701562119

Subproblem 2

n + -0.5 = -3.201562119 Simplifying n + -0.5 = -3.201562119 Reorder the terms: -0.5 + n = -3.201562119 Solving -0.5 + n = -3.201562119 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 = -3.201562119 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -3.201562119 + 0.5 n = -3.201562119 + 0.5 Combine like terms: -3.201562119 + 0.5 = -2.701562119 n = -2.701562119 Simplifying n = -2.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.701562119, -2.701562119}

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