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Simplifying -21 = 3r(r + -9) + -9 Reorder the terms: -21 = 3r(-9 + r) + -9 -21 = (-9 * 3r + r * 3r) + -9 -21 = (-27r + 3r2) + -9 Reorder the terms: -21 = -9 + -27r + 3r2 Solving -21 = -9 + -27r + 3r2 Solving for variable 'r'. Combine like terms: -21 + 9 = -12 -12 + 27r + -3r2 = -9 + -27r + 3r2 + 9 + 27r + -3r2 Reorder the terms: -12 + 27r + -3r2 = -9 + 9 + -27r + 27r + 3r2 + -3r2 Combine like terms: -9 + 9 = 0 -12 + 27r + -3r2 = 0 + -27r + 27r + 3r2 + -3r2 -12 + 27r + -3r2 = -27r + 27r + 3r2 + -3r2 Combine like terms: -27r + 27r = 0 -12 + 27r + -3r2 = 0 + 3r2 + -3r2 -12 + 27r + -3r2 = 3r2 + -3r2 Combine like terms: 3r2 + -3r2 = 0 -12 + 27r + -3r2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + 9r + -1r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + 9r + -1r2)' equal to zero and attempt to solve: Simplifying -4 + 9r + -1r2 = 0 Solving -4 + 9r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 4 + -9r + r2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -9r + -4 + r2 = 0 + -4 Reorder the terms: 4 + -4 + -9r + r2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -9r + r2 = 0 + -4 -9r + r2 = 0 + -4 Combine like terms: 0 + -4 = -4 -9r + r2 = -4 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 = -4 + 20.25 Reorder the terms: 20.25 + -9r + r2 = -4 + 20.25 Combine like terms: -4 + 20.25 = 16.25 20.25 + -9r + r2 = 16.25 Factor a perfect square on the left side: (r + -4.5)(r + -4.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (r + -4.5) equal to 4.031128874 and -4.031128874.Subproblem 1
r + -4.5 = 4.031128874 Simplifying r + -4.5 = 4.031128874 Reorder the terms: -4.5 + r = 4.031128874 Solving -4.5 + r = 4.031128874 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 = 4.031128874 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = 4.031128874 + 4.5 r = 4.031128874 + 4.5 Combine like terms: 4.031128874 + 4.5 = 8.531128874 r = 8.531128874 Simplifying r = 8.531128874Subproblem 2
r + -4.5 = -4.031128874 Simplifying r + -4.5 = -4.031128874 Reorder the terms: -4.5 + r = -4.031128874 Solving -4.5 + r = -4.031128874 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 = -4.031128874 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = -4.031128874 + 4.5 r = -4.031128874 + 4.5 Combine like terms: -4.031128874 + 4.5 = 0.468871126 r = 0.468871126 Simplifying r = 0.468871126Solution
The solution to the problem is based on the solutions from the subproblems. r = {8.531128874, 0.468871126}Solution
r = {8.531128874, 0.468871126}
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