r^2-5r+1=0

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Solution for r^2-5r+1=0 equation:


Simplifying
r2 + -5r + 1 = 0

Reorder the terms:
1 + -5r + r2 = 0

Solving
1 + -5r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5r + -1 + r2 = 0 + -1

Reorder the terms:
1 + -1 + -5r + r2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5r + r2 = 0 + -1
-5r + r2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5r + r2 = -1

The r term is -5r.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5r + 6.25 + r2 = -1 + 6.25

Reorder the terms:
6.25 + -5r + r2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5r + r2 = 5.25

Factor a perfect square on the left side:
(r + -2.5)(r + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(r + -2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

r + -2.5 = 2.291287847 Simplifying r + -2.5 = 2.291287847 Reorder the terms: -2.5 + r = 2.291287847 Solving -2.5 + r = 2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = 2.291287847 + 2.5 r = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 r = 4.791287847 Simplifying r = 4.791287847

Subproblem 2

r + -2.5 = -2.291287847 Simplifying r + -2.5 = -2.291287847 Reorder the terms: -2.5 + r = -2.291287847 Solving -2.5 + r = -2.291287847 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r = -2.291287847 + 2.5 r = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 r = 0.208712153 Simplifying r = 0.208712153

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.791287847, 0.208712153}

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