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Simplifying 6r3 + 42r2 + -18r = 0 Reorder the terms: -18r + 42r2 + 6r3 = 0 Solving -18r + 42r2 + 6r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '6r'. 6r(-3 + 7r + r2) = 0 Ignore the factor 6.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 '(-3 + 7r + r2)' equal to zero and attempt to solve: Simplifying -3 + 7r + r2 = 0 Solving -3 + 7r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 7r + 3 + r2 = 0 + 3 Reorder the terms: -3 + 3 + 7r + r2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 7r + r2 = 0 + 3 7r + r2 = 0 + 3 Combine like terms: 0 + 3 = 3 7r + r2 = 3 The r term is 7r. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7r + 12.25 + r2 = 3 + 12.25 Reorder the terms: 12.25 + 7r + r2 = 3 + 12.25 Combine like terms: 3 + 12.25 = 15.25 12.25 + 7r + r2 = 15.25 Factor a perfect square on the left side: (r + 3.5)(r + 3.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (r + 3.5) equal to 3.905124838 and -3.905124838.Subproblem 1
r + 3.5 = 3.905124838 Simplifying r + 3.5 = 3.905124838 Reorder the terms: 3.5 + r = 3.905124838 Solving 3.5 + r = 3.905124838 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + r = 3.905124838 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + r = 3.905124838 + -3.5 r = 3.905124838 + -3.5 Combine like terms: 3.905124838 + -3.5 = 0.405124838 r = 0.405124838 Simplifying r = 0.405124838Subproblem 2
r + 3.5 = -3.905124838 Simplifying r + 3.5 = -3.905124838 Reorder the terms: 3.5 + r = -3.905124838 Solving 3.5 + r = -3.905124838 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + r = -3.905124838 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + r = -3.905124838 + -3.5 r = -3.905124838 + -3.5 Combine like terms: -3.905124838 + -3.5 = -7.405124838 r = -7.405124838 Simplifying r = -7.405124838Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.405124838, -7.405124838}Solution
r = {0, 0.405124838, -7.405124838}
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