9*r^2-11*r+3=0

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Solution for 9*r^2-11*r+3=0 equation:


Simplifying
9r2 + -11r + 3 = 0

Reorder the terms:
3 + -11r + 9r2 = 0

Solving
3 + -11r + 9r2 = 0

Solving for variable 'r'.

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

Divide each side by '9'.
0.3333333333 + -1.222222222r + r2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -1.222222222r + -0.3333333333 + r2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -1.222222222r + r2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -1.222222222r + r2 = 0 + -0.3333333333
-1.222222222r + r2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-1.222222222r + r2 = -0.3333333333

The r term is -1.222222222r.  Take half its coefficient (-0.611111111).
Square it (0.3734567900) and add it to both sides.

Add '0.3734567900' to each side of the equation.
-1.222222222r + 0.3734567900 + r2 = -0.3333333333 + 0.3734567900

Reorder the terms:
0.3734567900 + -1.222222222r + r2 = -0.3333333333 + 0.3734567900

Combine like terms: -0.3333333333 + 0.3734567900 = 0.0401234567
0.3734567900 + -1.222222222r + r2 = 0.0401234567

Factor a perfect square on the left side:
(r + -0.611111111)(r + -0.611111111) = 0.0401234567

Calculate the square root of the right side: 0.200308404

Break this problem into two subproblems by setting 
(r + -0.611111111) equal to 0.200308404 and -0.200308404.

Subproblem 1

r + -0.611111111 = 0.200308404 Simplifying r + -0.611111111 = 0.200308404 Reorder the terms: -0.611111111 + r = 0.200308404 Solving -0.611111111 + r = 0.200308404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = 0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = 0.200308404 + 0.611111111 r = 0.200308404 + 0.611111111 Combine like terms: 0.200308404 + 0.611111111 = 0.811419515 r = 0.811419515 Simplifying r = 0.811419515

Subproblem 2

r + -0.611111111 = -0.200308404 Simplifying r + -0.611111111 = -0.200308404 Reorder the terms: -0.611111111 + r = -0.200308404 Solving -0.611111111 + r = -0.200308404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = -0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = -0.200308404 + 0.611111111 r = -0.200308404 + 0.611111111 Combine like terms: -0.200308404 + 0.611111111 = 0.410802707 r = 0.410802707 Simplifying r = 0.410802707

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.811419515, 0.410802707}

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