r^2-20r+41=0

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Solution for r^2-20r+41=0 equation:


Simplifying
r2 + -20r + 41 = 0

Reorder the terms:
41 + -20r + r2 = 0

Solving
41 + -20r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-41' to each side of the equation.
41 + -20r + -41 + r2 = 0 + -41

Reorder the terms:
41 + -41 + -20r + r2 = 0 + -41

Combine like terms: 41 + -41 = 0
0 + -20r + r2 = 0 + -41
-20r + r2 = 0 + -41

Combine like terms: 0 + -41 = -41
-20r + r2 = -41

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

Add '100' to each side of the equation.
-20r + 100 + r2 = -41 + 100

Reorder the terms:
100 + -20r + r2 = -41 + 100

Combine like terms: -41 + 100 = 59
100 + -20r + r2 = 59

Factor a perfect square on the left side:
(r + -10)(r + -10) = 59

Calculate the square root of the right side: 7.681145748

Break this problem into two subproblems by setting 
(r + -10) equal to 7.681145748 and -7.681145748.

Subproblem 1

r + -10 = 7.681145748 Simplifying r + -10 = 7.681145748 Reorder the terms: -10 + r = 7.681145748 Solving -10 + r = 7.681145748 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + r = 7.681145748 + 10 Combine like terms: -10 + 10 = 0 0 + r = 7.681145748 + 10 r = 7.681145748 + 10 Combine like terms: 7.681145748 + 10 = 17.681145748 r = 17.681145748 Simplifying r = 17.681145748

Subproblem 2

r + -10 = -7.681145748 Simplifying r + -10 = -7.681145748 Reorder the terms: -10 + r = -7.681145748 Solving -10 + r = -7.681145748 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + r = -7.681145748 + 10 Combine like terms: -10 + 10 = 0 0 + r = -7.681145748 + 10 r = -7.681145748 + 10 Combine like terms: -7.681145748 + 10 = 2.318854252 r = 2.318854252 Simplifying r = 2.318854252

Solution

The solution to the problem is based on the solutions from the subproblems. r = {17.681145748, 2.318854252}

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