n^2-20n+100=40

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Solution for n^2-20n+100=40 equation:


Simplifying
n2 + -20n + 100 = 40

Reorder the terms:
100 + -20n + n2 = 40

Solving
100 + -20n + n2 = 40

Solving for variable 'n'.

Reorder the terms:
100 + -40 + -20n + n2 = 40 + -40

Combine like terms: 100 + -40 = 60
60 + -20n + n2 = 40 + -40

Combine like terms: 40 + -40 = 0
60 + -20n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-60' to each side of the equation.
60 + -20n + -60 + n2 = 0 + -60

Reorder the terms:
60 + -60 + -20n + n2 = 0 + -60

Combine like terms: 60 + -60 = 0
0 + -20n + n2 = 0 + -60
-20n + n2 = 0 + -60

Combine like terms: 0 + -60 = -60
-20n + n2 = -60

The n term is -20n.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20n + 100 + n2 = -60 + 100

Reorder the terms:
100 + -20n + n2 = -60 + 100

Combine like terms: -60 + 100 = 40
100 + -20n + n2 = 40

Factor a perfect square on the left side:
(n + -10)(n + -10) = 40

Calculate the square root of the right side: 6.32455532

Break this problem into two subproblems by setting 
(n + -10) equal to 6.32455532 and -6.32455532.

Subproblem 1

n + -10 = 6.32455532 Simplifying n + -10 = 6.32455532 Reorder the terms: -10 + n = 6.32455532 Solving -10 + n = 6.32455532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + n = 6.32455532 + 10 Combine like terms: -10 + 10 = 0 0 + n = 6.32455532 + 10 n = 6.32455532 + 10 Combine like terms: 6.32455532 + 10 = 16.32455532 n = 16.32455532 Simplifying n = 16.32455532

Subproblem 2

n + -10 = -6.32455532 Simplifying n + -10 = -6.32455532 Reorder the terms: -10 + n = -6.32455532 Solving -10 + n = -6.32455532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + n = -6.32455532 + 10 Combine like terms: -10 + 10 = 0 0 + n = -6.32455532 + 10 n = -6.32455532 + 10 Combine like terms: -6.32455532 + 10 = 3.67544468 n = 3.67544468 Simplifying n = 3.67544468

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.32455532, 3.67544468}

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