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Simplifying 3r3 + 15r2 + -54r = 0 Reorder the terms: -54r + 15r2 + 3r3 = 0 Solving -54r + 15r2 + 3r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3r'. 3r(-18 + 5r + r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r = 0Subproblem 2
Set the factor '(-18 + 5r + r2)' equal to zero and attempt to solve: Simplifying -18 + 5r + r2 = 0 Solving -18 + 5r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 5r + 18 + r2 = 0 + 18 Reorder the terms: -18 + 18 + 5r + r2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 5r + r2 = 0 + 18 5r + r2 = 0 + 18 Combine like terms: 0 + 18 = 18 5r + r2 = 18 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = 18 + 6.25 Reorder the terms: 6.25 + 5r + r2 = 18 + 6.25 Combine like terms: 18 + 6.25 = 24.25 6.25 + 5r + r2 = 24.25 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (r + 2.5) equal to 4.924428901 and -4.924428901.Subproblem 1
r + 2.5 = 4.924428901 Simplifying r + 2.5 = 4.924428901 Reorder the terms: 2.5 + r = 4.924428901 Solving 2.5 + r = 4.924428901 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 4.924428901 + -2.5 r = 4.924428901 + -2.5 Combine like terms: 4.924428901 + -2.5 = 2.424428901 r = 2.424428901 Simplifying r = 2.424428901Subproblem 2
r + 2.5 = -4.924428901 Simplifying r + 2.5 = -4.924428901 Reorder the terms: 2.5 + r = -4.924428901 Solving 2.5 + r = -4.924428901 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -4.924428901 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -4.924428901 + -2.5 r = -4.924428901 + -2.5 Combine like terms: -4.924428901 + -2.5 = -7.424428901 r = -7.424428901 Simplifying r = -7.424428901Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.424428901, -7.424428901}Solution
r = {0, 2.424428901, -7.424428901}
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