r=4-9r^2

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Solution for r=4-9r^2 equation:


Simplifying
r = 4 + -9r2

Solving
r = 4 + -9r2

Solving for variable 'r'.

Reorder the terms:
-4 + r + 9r2 = 4 + -9r2 + -4 + 9r2

Reorder the terms:
-4 + r + 9r2 = 4 + -4 + -9r2 + 9r2

Combine like terms: 4 + -4 = 0
-4 + r + 9r2 = 0 + -9r2 + 9r2
-4 + r + 9r2 = -9r2 + 9r2

Combine like terms: -9r2 + 9r2 = 0
-4 + r + 9r2 = 0

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

Divide each side by '9'.
-0.4444444444 + 0.1111111111r + r2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 0.1111111111r + 0.4444444444 + r2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 0.1111111111r + r2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 0.1111111111r + r2 = 0 + 0.4444444444
0.1111111111r + r2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
0.1111111111r + r2 = 0.4444444444

The r term is r.  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.
0.1111111111r + 0.25 + r2 = 0.4444444444 + 0.25

Reorder the terms:
0.25 + 0.1111111111r + r2 = 0.4444444444 + 0.25

Combine like terms: 0.4444444444 + 0.25 = 0.6944444444
0.25 + 0.1111111111r + r2 = 0.6944444444

Factor a perfect square on the left side:
(r + 0.5)(r + 0.5) = 0.6944444444

Calculate the square root of the right side: 0.833333333

Break this problem into two subproblems by setting 
(r + 0.5) equal to 0.833333333 and -0.833333333.

Subproblem 1

r + 0.5 = 0.833333333 Simplifying r + 0.5 = 0.833333333 Reorder the terms: 0.5 + r = 0.833333333 Solving 0.5 + r = 0.833333333 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 = 0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.833333333 + -0.5 r = 0.833333333 + -0.5 Combine like terms: 0.833333333 + -0.5 = 0.333333333 r = 0.333333333 Simplifying r = 0.333333333

Subproblem 2

r + 0.5 = -0.833333333 Simplifying r + 0.5 = -0.833333333 Reorder the terms: 0.5 + r = -0.833333333 Solving 0.5 + r = -0.833333333 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 = -0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.833333333 + -0.5 r = -0.833333333 + -0.5 Combine like terms: -0.833333333 + -0.5 = -1.333333333 r = -1.333333333 Simplifying r = -1.333333333

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.333333333, -1.333333333}

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