n^2-20n+61=4

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Solution for n^2-20n+61=4 equation:


Simplifying
n2 + -20n + 61 = 4

Reorder the terms:
61 + -20n + n2 = 4

Solving
61 + -20n + n2 = 4

Solving for variable 'n'.

Reorder the terms:
61 + -4 + -20n + n2 = 4 + -4

Combine like terms: 61 + -4 = 57
57 + -20n + n2 = 4 + -4

Combine like terms: 4 + -4 = 0
57 + -20n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-57' to each side of the equation.
57 + -20n + -57 + n2 = 0 + -57

Reorder the terms:
57 + -57 + -20n + n2 = 0 + -57

Combine like terms: 57 + -57 = 0
0 + -20n + n2 = 0 + -57
-20n + n2 = 0 + -57

Combine like terms: 0 + -57 = -57
-20n + n2 = -57

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 = -57 + 100

Reorder the terms:
100 + -20n + n2 = -57 + 100

Combine like terms: -57 + 100 = 43
100 + -20n + n2 = 43

Factor a perfect square on the left side:
(n + -10)(n + -10) = 43

Calculate the square root of the right side: 6.557438524

Break this problem into two subproblems by setting 
(n + -10) equal to 6.557438524 and -6.557438524.

Subproblem 1

n + -10 = 6.557438524 Simplifying n + -10 = 6.557438524 Reorder the terms: -10 + n = 6.557438524 Solving -10 + n = 6.557438524 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 = 6.557438524 + 10 Combine like terms: -10 + 10 = 0 0 + n = 6.557438524 + 10 n = 6.557438524 + 10 Combine like terms: 6.557438524 + 10 = 16.557438524 n = 16.557438524 Simplifying n = 16.557438524

Subproblem 2

n + -10 = -6.557438524 Simplifying n + -10 = -6.557438524 Reorder the terms: -10 + n = -6.557438524 Solving -10 + n = -6.557438524 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 = -6.557438524 + 10 Combine like terms: -10 + 10 = 0 0 + n = -6.557438524 + 10 n = -6.557438524 + 10 Combine like terms: -6.557438524 + 10 = 3.442561476 n = 3.442561476 Simplifying n = 3.442561476

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.557438524, 3.442561476}

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