-21=3r(r-9)-10

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Solution for -21=3r(r-9)-10 equation:


Simplifying
-21 = 3r(r + -9) + -10

Reorder the terms:
-21 = 3r(-9 + r) + -10
-21 = (-9 * 3r + r * 3r) + -10
-21 = (-27r + 3r2) + -10

Reorder the terms:
-21 = -10 + -27r + 3r2

Solving
-21 = -10 + -27r + 3r2

Solving for variable 'r'.

Combine like terms: -21 + 10 = -11
-11 + 27r + -3r2 = -10 + -27r + 3r2 + 10 + 27r + -3r2

Reorder the terms:
-11 + 27r + -3r2 = -10 + 10 + -27r + 27r + 3r2 + -3r2

Combine like terms: -10 + 10 = 0
-11 + 27r + -3r2 = 0 + -27r + 27r + 3r2 + -3r2
-11 + 27r + -3r2 = -27r + 27r + 3r2 + -3r2

Combine like terms: -27r + 27r = 0
-11 + 27r + -3r2 = 0 + 3r2 + -3r2
-11 + 27r + -3r2 = 3r2 + -3r2

Combine like terms: 3r2 + -3r2 = 0
-11 + 27r + -3r2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
3.666666667 + -9r + r2 = 0

Move the constant term to the right:

Add '-3.666666667' to each side of the equation.
3.666666667 + -9r + -3.666666667 + r2 = 0 + -3.666666667

Reorder the terms:
3.666666667 + -3.666666667 + -9r + r2 = 0 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + -9r + r2 = 0 + -3.666666667
-9r + r2 = 0 + -3.666666667

Combine like terms: 0 + -3.666666667 = -3.666666667
-9r + r2 = -3.666666667

The r term is -9r.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9r + 20.25 + r2 = -3.666666667 + 20.25

Reorder the terms:
20.25 + -9r + r2 = -3.666666667 + 20.25

Combine like terms: -3.666666667 + 20.25 = 16.583333333
20.25 + -9r + r2 = 16.583333333

Factor a perfect square on the left side:
(r + -4.5)(r + -4.5) = 16.583333333

Calculate the square root of the right side: 4.072263908

Break this problem into two subproblems by setting 
(r + -4.5) equal to 4.072263908 and -4.072263908.

Subproblem 1

r + -4.5 = 4.072263908 Simplifying r + -4.5 = 4.072263908 Reorder the terms: -4.5 + r = 4.072263908 Solving -4.5 + r = 4.072263908 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = 4.072263908 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = 4.072263908 + 4.5 r = 4.072263908 + 4.5 Combine like terms: 4.072263908 + 4.5 = 8.572263908 r = 8.572263908 Simplifying r = 8.572263908

Subproblem 2

r + -4.5 = -4.072263908 Simplifying r + -4.5 = -4.072263908 Reorder the terms: -4.5 + r = -4.072263908 Solving -4.5 + r = -4.072263908 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = -4.072263908 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = -4.072263908 + 4.5 r = -4.072263908 + 4.5 Combine like terms: -4.072263908 + 4.5 = 0.427736092 r = 0.427736092 Simplifying r = 0.427736092

Solution

The solution to the problem is based on the solutions from the subproblems. r = {8.572263908, 0.427736092}

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