75=0.05r(r+2)

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Solution for 75=0.05r(r+2) equation:


Simplifying
75 = 0.05r(r + 2)

Reorder the terms:
75 = 0.05r(2 + r)
75 = (2 * 0.05r + r * 0.05r)
75 = (0.1r + 0.05r2)

Solving
75 = 0.1r + 0.05r2

Solving for variable 'r'.

Reorder the terms:
75 + -0.1r + -0.05r2 = 0.1r + -0.1r + 0.05r2 + -0.05r2

Combine like terms: 0.1r + -0.1r = 0.0
75 + -0.1r + -0.05r2 = 0.0 + 0.05r2 + -0.05r2
75 + -0.1r + -0.05r2 = 0.05r2 + -0.05r2

Combine like terms: 0.05r2 + -0.05r2 = 0.00
75 + -0.1r + -0.05r2 = 0.00

Begin completing the square.  Divide all terms by
-0.05 the coefficient of the squared term: 

Divide each side by '-0.05'.
-1500 + 2r + r2 = 0

Move the constant term to the right:

Add '1500' to each side of the equation.
-1500 + 2r + 1500 + r2 = 0 + 1500

Reorder the terms:
-1500 + 1500 + 2r + r2 = 0 + 1500

Combine like terms: -1500 + 1500 = 0
0 + 2r + r2 = 0 + 1500
2r + r2 = 0 + 1500

Combine like terms: 0 + 1500 = 1500
2r + r2 = 1500

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 = 1500 + 1

Reorder the terms:
1 + 2r + r2 = 1500 + 1

Combine like terms: 1500 + 1 = 1501
1 + 2r + r2 = 1501

Factor a perfect square on the left side:
(r + 1)(r + 1) = 1501

Calculate the square root of the right side: 38.742741256

Break this problem into two subproblems by setting 
(r + 1) equal to 38.742741256 and -38.742741256.

Subproblem 1

r + 1 = 38.742741256 Simplifying r + 1 = 38.742741256 Reorder the terms: 1 + r = 38.742741256 Solving 1 + r = 38.742741256 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 = 38.742741256 + -1 Combine like terms: 1 + -1 = 0 0 + r = 38.742741256 + -1 r = 38.742741256 + -1 Combine like terms: 38.742741256 + -1 = 37.742741256 r = 37.742741256 Simplifying r = 37.742741256

Subproblem 2

r + 1 = -38.742741256 Simplifying r + 1 = -38.742741256 Reorder the terms: 1 + r = -38.742741256 Solving 1 + r = -38.742741256 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 = -38.742741256 + -1 Combine like terms: 1 + -1 = 0 0 + r = -38.742741256 + -1 r = -38.742741256 + -1 Combine like terms: -38.742741256 + -1 = -39.742741256 r = -39.742741256 Simplifying r = -39.742741256

Solution

The solution to the problem is based on the solutions from the subproblems. r = {37.742741256, -39.742741256}

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