49.5=r(15-r)

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Solution for 49.5=r(15-r) equation:


Simplifying
49.5 = r(15 + -1r)
49.5 = (15 * r + -1r * r)
49.5 = (15r + -1r2)

Solving
49.5 = 15r + -1r2

Solving for variable 'r'.

Reorder the terms:
49.5 + -15r + r2 = 15r + -15r + -1r2 + r2

Combine like terms: 15r + -15r = 0
49.5 + -15r + r2 = 0 + -1r2 + r2
49.5 + -15r + r2 = -1r2 + r2

Combine like terms: -1r2 + r2 = 0
49.5 + -15r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-49.5' to each side of the equation.
49.5 + -15r + -49.5 + r2 = 0 + -49.5

Reorder the terms:
49.5 + -49.5 + -15r + r2 = 0 + -49.5

Combine like terms: 49.5 + -49.5 = 0.0
0.0 + -15r + r2 = 0 + -49.5
-15r + r2 = 0 + -49.5

Combine like terms: 0 + -49.5 = -49.5
-15r + r2 = -49.5

The r term is -15r.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15r + 56.25 + r2 = -49.5 + 56.25

Reorder the terms:
56.25 + -15r + r2 = -49.5 + 56.25

Combine like terms: -49.5 + 56.25 = 6.75
56.25 + -15r + r2 = 6.75

Factor a perfect square on the left side:
(r + -7.5)(r + -7.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(r + -7.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

r + -7.5 = 2.598076211 Simplifying r + -7.5 = 2.598076211 Reorder the terms: -7.5 + r = 2.598076211 Solving -7.5 + r = 2.598076211 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + r = 2.598076211 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + r = 2.598076211 + 7.5 r = 2.598076211 + 7.5 Combine like terms: 2.598076211 + 7.5 = 10.098076211 r = 10.098076211 Simplifying r = 10.098076211

Subproblem 2

r + -7.5 = -2.598076211 Simplifying r + -7.5 = -2.598076211 Reorder the terms: -7.5 + r = -2.598076211 Solving -7.5 + r = -2.598076211 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + r = -2.598076211 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + r = -2.598076211 + 7.5 r = -2.598076211 + 7.5 Combine like terms: -2.598076211 + 7.5 = 4.901923789 r = 4.901923789 Simplifying r = 4.901923789

Solution

The solution to the problem is based on the solutions from the subproblems. r = {10.098076211, 4.901923789}

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