4r^2-104r+5=0

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Solution for 4r^2-104r+5=0 equation:


Simplifying
4r2 + -104r + 5 = 0

Reorder the terms:
5 + -104r + 4r2 = 0

Solving
5 + -104r + 4r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
1.25 + -26r + r2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + -26r + -1.25 + r2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + -26r + r2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + -26r + r2 = 0 + -1.25
-26r + r2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
-26r + r2 = -1.25

The r term is -26r.  Take half its coefficient (-13).
Square it (169) and add it to both sides.

Add '169' to each side of the equation.
-26r + 169 + r2 = -1.25 + 169

Reorder the terms:
169 + -26r + r2 = -1.25 + 169

Combine like terms: -1.25 + 169 = 167.75
169 + -26r + r2 = 167.75

Factor a perfect square on the left side:
(r + -13)(r + -13) = 167.75

Calculate the square root of the right side: 12.951833847

Break this problem into two subproblems by setting 
(r + -13) equal to 12.951833847 and -12.951833847.

Subproblem 1

r + -13 = 12.951833847 Simplifying r + -13 = 12.951833847 Reorder the terms: -13 + r = 12.951833847 Solving -13 + r = 12.951833847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + r = 12.951833847 + 13 Combine like terms: -13 + 13 = 0 0 + r = 12.951833847 + 13 r = 12.951833847 + 13 Combine like terms: 12.951833847 + 13 = 25.951833847 r = 25.951833847 Simplifying r = 25.951833847

Subproblem 2

r + -13 = -12.951833847 Simplifying r + -13 = -12.951833847 Reorder the terms: -13 + r = -12.951833847 Solving -13 + r = -12.951833847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + r = -12.951833847 + 13 Combine like terms: -13 + 13 = 0 0 + r = -12.951833847 + 13 r = -12.951833847 + 13 Combine like terms: -12.951833847 + 13 = 0.048166153 r = 0.048166153 Simplifying r = 0.048166153

Solution

The solution to the problem is based on the solutions from the subproblems. r = {25.951833847, 0.048166153}

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