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Simplifying 4r2 + -6r + 2r2 + -2 = 0 Reorder the terms: -2 + -6r + 4r2 + 2r2 = 0 Combine like terms: 4r2 + 2r2 = 6r2 -2 + -6r + 6r2 = 0 Solving -2 + -6r + 6r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -3r + 3r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -3r + 3r2)' equal to zero and attempt to solve: Simplifying -1 + -3r + 3r2 = 0 Solving -1 + -3r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -1r + r2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1r + 0.3333333333 + r2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1r + r2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1r + r2 = 0 + 0.3333333333 -1r + r2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1r + r2 = 0.3333333333 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 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + -1r + r2 = 0.5833333333 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (r + -0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
r + -0.5 = 0.763762616 Simplifying r + -0.5 = 0.763762616 Reorder the terms: -0.5 + r = 0.763762616 Solving -0.5 + r = 0.763762616 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 = 0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 0.763762616 + 0.5 r = 0.763762616 + 0.5 Combine like terms: 0.763762616 + 0.5 = 1.263762616 r = 1.263762616 Simplifying r = 1.263762616Subproblem 2
r + -0.5 = -0.763762616 Simplifying r + -0.5 = -0.763762616 Reorder the terms: -0.5 + r = -0.763762616 Solving -0.5 + r = -0.763762616 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 = -0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -0.763762616 + 0.5 r = -0.763762616 + 0.5 Combine like terms: -0.763762616 + 0.5 = -0.263762616 r = -0.263762616 Simplifying r = -0.263762616Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.263762616, -0.263762616}Solution
r = {1.263762616, -0.263762616}
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