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Simplifying 6r3 + 15r2 + -6r = 0 Reorder the terms: -6r + 15r2 + 6r3 = 0 Solving -6r + 15r2 + 6r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3r'. 3r(-2 + 5r + 2r2) = 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 '(-2 + 5r + 2r2)' equal to zero and attempt to solve: Simplifying -2 + 5r + 2r2 = 0 Solving -2 + 5r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1 + 2.5r + r2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.5r + 1 + r2 = 0 + 1 Reorder the terms: -1 + 1 + 2.5r + r2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.5r + r2 = 0 + 1 2.5r + r2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.5r + r2 = 1 The r term is 2.5r. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5r + 1.5625 + r2 = 1 + 1.5625 Reorder the terms: 1.5625 + 2.5r + r2 = 1 + 1.5625 Combine like terms: 1 + 1.5625 = 2.5625 1.5625 + 2.5r + r2 = 2.5625 Factor a perfect square on the left side: (r + 1.25)(r + 1.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (r + 1.25) equal to 1.600781059 and -1.600781059.Subproblem 1
r + 1.25 = 1.600781059 Simplifying r + 1.25 = 1.600781059 Reorder the terms: 1.25 + r = 1.600781059 Solving 1.25 + r = 1.600781059 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = 1.600781059 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = 1.600781059 + -1.25 r = 1.600781059 + -1.25 Combine like terms: 1.600781059 + -1.25 = 0.350781059 r = 0.350781059 Simplifying r = 0.350781059Subproblem 2
r + 1.25 = -1.600781059 Simplifying r + 1.25 = -1.600781059 Reorder the terms: 1.25 + r = -1.600781059 Solving 1.25 + r = -1.600781059 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + r = -1.600781059 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + r = -1.600781059 + -1.25 r = -1.600781059 + -1.25 Combine like terms: -1.600781059 + -1.25 = -2.850781059 r = -2.850781059 Simplifying r = -2.850781059Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.350781059, -2.850781059}Solution
r = {0, 0.350781059, -2.850781059}
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