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Simplifying 10r2 + 3r + -13r = 5 Reorder the terms: 3r + -13r + 10r2 = 5 Combine like terms: 3r + -13r = -10r -10r + 10r2 = 5 Solving -10r + 10r2 = 5 Solving for variable 'r'. Reorder the terms: -5 + -10r + 10r2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -10r + 10r2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-1 + -2r + 2r2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-1 + -2r + 2r2)' equal to zero and attempt to solve: Simplifying -1 + -2r + 2r2 = 0 Solving -1 + -2r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -1r + r2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1r + 0.5 + r2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1r + r2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1r + r2 = 0 + 0.5 -1r + r2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1r + r2 = 0.5 The r term is -1r. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1r + 0.25 + r2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1r + r2 = 0.75 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (r + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
r + -0.5 = 0.866025404 Simplifying r + -0.5 = 0.866025404 Reorder the terms: -0.5 + r = 0.866025404 Solving -0.5 + r = 0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 0.866025404 + 0.5 r = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 r = 1.366025404 Simplifying r = 1.366025404Subproblem 2
r + -0.5 = -0.866025404 Simplifying r + -0.5 = -0.866025404 Reorder the terms: -0.5 + r = -0.866025404 Solving -0.5 + r = -0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -0.866025404 + 0.5 r = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 r = -0.366025404 Simplifying r = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.366025404, -0.366025404}Solution
r = {1.366025404, -0.366025404}
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