20r^2=20r+1

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Solution for 20r^2=20r+1 equation:


Simplifying
20r2 = 20r + 1

Reorder the terms:
20r2 = 1 + 20r

Solving
20r2 = 1 + 20r

Solving for variable 'r'.

Reorder the terms:
-1 + -20r + 20r2 = 1 + 20r + -1 + -20r

Reorder the terms:
-1 + -20r + 20r2 = 1 + -1 + 20r + -20r

Combine like terms: 1 + -1 = 0
-1 + -20r + 20r2 = 0 + 20r + -20r
-1 + -20r + 20r2 = 20r + -20r

Combine like terms: 20r + -20r = 0
-1 + -20r + 20r2 = 0

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

Divide each side by '20'.
-0.05 + -1r + r2 = 0

Move the constant term to the right:

Add '0.05' to each side of the equation.
-0.05 + -1r + 0.05 + r2 = 0 + 0.05

Reorder the terms:
-0.05 + 0.05 + -1r + r2 = 0 + 0.05

Combine like terms: -0.05 + 0.05 = 0.00
0.00 + -1r + r2 = 0 + 0.05
-1r + r2 = 0 + 0.05

Combine like terms: 0 + 0.05 = 0.05
-1r + r2 = 0.05

The r term is -1r.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1r + 0.25 + r2 = 0.05 + 0.25

Reorder the terms:
0.25 + -1r + r2 = 0.05 + 0.25

Combine like terms: 0.05 + 0.25 = 0.3
0.25 + -1r + r2 = 0.3

Factor a perfect square on the left side:
(r + -0.5)(r + -0.5) = 0.3

Calculate the square root of the right side: 0.547722558

Break this problem into two subproblems by setting 
(r + -0.5) equal to 0.547722558 and -0.547722558.

Subproblem 1

r + -0.5 = 0.547722558 Simplifying r + -0.5 = 0.547722558 Reorder the terms: -0.5 + r = 0.547722558 Solving -0.5 + r = 0.547722558 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = 0.547722558 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 0.547722558 + 0.5 r = 0.547722558 + 0.5 Combine like terms: 0.547722558 + 0.5 = 1.047722558 r = 1.047722558 Simplifying r = 1.047722558

Subproblem 2

r + -0.5 = -0.547722558 Simplifying r + -0.5 = -0.547722558 Reorder the terms: -0.5 + r = -0.547722558 Solving -0.5 + r = -0.547722558 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = -0.547722558 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -0.547722558 + 0.5 r = -0.547722558 + 0.5 Combine like terms: -0.547722558 + 0.5 = -0.047722558 r = -0.047722558 Simplifying r = -0.047722558

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.047722558, -0.047722558}

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