400=r*(12+r)

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Solution for 400=r*(12+r) equation:


Simplifying
400 = r(12 + r)
400 = (12 * r + r * r)
400 = (12r + r2)

Solving
400 = 12r + r2

Solving for variable 'r'.

Reorder the terms:
400 + -12r + -1r2 = 12r + -12r + r2 + -1r2

Combine like terms: 12r + -12r = 0
400 + -12r + -1r2 = 0 + r2 + -1r2
400 + -12r + -1r2 = r2 + -1r2

Combine like terms: r2 + -1r2 = 0
400 + -12r + -1r2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-400 + 12r + r2 = 0

Move the constant term to the right:

Add '400' to each side of the equation.
-400 + 12r + 400 + r2 = 0 + 400

Reorder the terms:
-400 + 400 + 12r + r2 = 0 + 400

Combine like terms: -400 + 400 = 0
0 + 12r + r2 = 0 + 400
12r + r2 = 0 + 400

Combine like terms: 0 + 400 = 400
12r + r2 = 400

The r term is 12r.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12r + 36 + r2 = 400 + 36

Reorder the terms:
36 + 12r + r2 = 400 + 36

Combine like terms: 400 + 36 = 436
36 + 12r + r2 = 436

Factor a perfect square on the left side:
(r + 6)(r + 6) = 436

Calculate the square root of the right side: 20.880613018

Break this problem into two subproblems by setting 
(r + 6) equal to 20.880613018 and -20.880613018.

Subproblem 1

r + 6 = 20.880613018 Simplifying r + 6 = 20.880613018 Reorder the terms: 6 + r = 20.880613018 Solving 6 + r = 20.880613018 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = 20.880613018 + -6 Combine like terms: 6 + -6 = 0 0 + r = 20.880613018 + -6 r = 20.880613018 + -6 Combine like terms: 20.880613018 + -6 = 14.880613018 r = 14.880613018 Simplifying r = 14.880613018

Subproblem 2

r + 6 = -20.880613018 Simplifying r + 6 = -20.880613018 Reorder the terms: 6 + r = -20.880613018 Solving 6 + r = -20.880613018 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = -20.880613018 + -6 Combine like terms: 6 + -6 = 0 0 + r = -20.880613018 + -6 r = -20.880613018 + -6 Combine like terms: -20.880613018 + -6 = -26.880613018 r = -26.880613018 Simplifying r = -26.880613018

Solution

The solution to the problem is based on the solutions from the subproblems. r = {14.880613018, -26.880613018}

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