4r^2-4r+1=6

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Solution for 4r^2-4r+1=6 equation:


Simplifying
4r2 + -4r + 1 = 6

Reorder the terms:
1 + -4r + 4r2 = 6

Solving
1 + -4r + 4r2 = 6

Solving for variable 'r'.

Reorder the terms:
1 + -6 + -4r + 4r2 = 6 + -6

Combine like terms: 1 + -6 = -5
-5 + -4r + 4r2 = 6 + -6

Combine like terms: 6 + -6 = 0
-5 + -4r + 4r2 = 0

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

Divide each side by '4'.
-1.25 + -1r + r2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -1r + 1.25 + r2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -1r + r2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -1r + r2 = 0 + 1.25
-1r + r2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-1r + r2 = 1.25

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 = 1.25 + 0.25

Reorder the terms:
0.25 + -1r + r2 = 1.25 + 0.25

Combine like terms: 1.25 + 0.25 = 1.5
0.25 + -1r + r2 = 1.5

Factor a perfect square on the left side:
(r + -0.5)(r + -0.5) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(r + -0.5) equal to 1.224744871 and -1.224744871.

Subproblem 1

r + -0.5 = 1.224744871 Simplifying r + -0.5 = 1.224744871 Reorder the terms: -0.5 + r = 1.224744871 Solving -0.5 + r = 1.224744871 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 = 1.224744871 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 1.224744871 + 0.5 r = 1.224744871 + 0.5 Combine like terms: 1.224744871 + 0.5 = 1.724744871 r = 1.724744871 Simplifying r = 1.724744871

Subproblem 2

r + -0.5 = -1.224744871 Simplifying r + -0.5 = -1.224744871 Reorder the terms: -0.5 + r = -1.224744871 Solving -0.5 + r = -1.224744871 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 = -1.224744871 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -1.224744871 + 0.5 r = -1.224744871 + 0.5 Combine like terms: -1.224744871 + 0.5 = -0.724744871 r = -0.724744871 Simplifying r = -0.724744871

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.724744871, -0.724744871}

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