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