4r^2=8r-1

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Solution for 4r^2=8r-1 equation:


Simplifying
4r2 = 8r + -1

Reorder the terms:
4r2 = -1 + 8r

Solving
4r2 = -1 + 8r

Solving for variable 'r'.

Reorder the terms:
1 + -8r + 4r2 = -1 + 8r + 1 + -8r

Reorder the terms:
1 + -8r + 4r2 = -1 + 1 + 8r + -8r

Combine like terms: -1 + 1 = 0
1 + -8r + 4r2 = 0 + 8r + -8r
1 + -8r + 4r2 = 8r + -8r

Combine like terms: 8r + -8r = 0
1 + -8r + 4r2 = 0

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

Divide each side by '4'.
0.25 + -2r + r2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -2r + -0.25 + r2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -2r + r2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -2r + r2 = 0 + -0.25
-2r + r2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-2r + r2 = -0.25

The r term is -2r.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2r + 1 + r2 = -0.25 + 1

Reorder the terms:
1 + -2r + r2 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + -2r + r2 = 0.75

Factor a perfect square on the left side:
(r + -1)(r + -1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(r + -1) equal to 0.866025404 and -0.866025404.

Subproblem 1

r + -1 = 0.866025404 Simplifying r + -1 = 0.866025404 Reorder the terms: -1 + r = 0.866025404 Solving -1 + r = 0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + r = 0.866025404 + 1 r = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 r = 1.866025404 Simplifying r = 1.866025404

Subproblem 2

r + -1 = -0.866025404 Simplifying r + -1 = -0.866025404 Reorder the terms: -1 + r = -0.866025404 Solving -1 + r = -0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + r = -0.866025404 + 1 r = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 r = 0.133974596 Simplifying r = 0.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.866025404, 0.133974596}

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