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Simplifying 2r3 + -13r2 + -20r = 0 Reorder the terms: -20r + -13r2 + 2r3 = 0 Solving -20r + -13r2 + 2r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r'. r(-20 + -13r + 2r2) = 0Subproblem 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 '(-20 + -13r + 2r2)' equal to zero and attempt to solve: Simplifying -20 + -13r + 2r2 = 0 Solving -20 + -13r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10 + -6.5r + r2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -6.5r + 10 + r2 = 0 + 10 Reorder the terms: -10 + 10 + -6.5r + r2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -6.5r + r2 = 0 + 10 -6.5r + r2 = 0 + 10 Combine like terms: 0 + 10 = 10 -6.5r + r2 = 10 The r term is -6.5r. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5r + 10.5625 + r2 = 10 + 10.5625 Reorder the terms: 10.5625 + -6.5r + r2 = 10 + 10.5625 Combine like terms: 10 + 10.5625 = 20.5625 10.5625 + -6.5r + r2 = 20.5625 Factor a perfect square on the left side: (r + -3.25)(r + -3.25) = 20.5625 Calculate the square root of the right side: 4.534589287 Break this problem into two subproblems by setting (r + -3.25) equal to 4.534589287 and -4.534589287.Subproblem 1
r + -3.25 = 4.534589287 Simplifying r + -3.25 = 4.534589287 Reorder the terms: -3.25 + r = 4.534589287 Solving -3.25 + r = 4.534589287 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = 4.534589287 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = 4.534589287 + 3.25 r = 4.534589287 + 3.25 Combine like terms: 4.534589287 + 3.25 = 7.784589287 r = 7.784589287 Simplifying r = 7.784589287Subproblem 2
r + -3.25 = -4.534589287 Simplifying r + -3.25 = -4.534589287 Reorder the terms: -3.25 + r = -4.534589287 Solving -3.25 + r = -4.534589287 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = -4.534589287 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = -4.534589287 + 3.25 r = -4.534589287 + 3.25 Combine like terms: -4.534589287 + 3.25 = -1.284589287 r = -1.284589287 Simplifying r = -1.284589287Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.784589287, -1.284589287}Solution
r = {0, 7.784589287, -1.284589287}
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