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Simplifying 6r2 + -12r = 6 Reorder the terms: -12r + 6r2 = 6 Solving -12r + 6r2 = 6 Solving for variable 'r'. Reorder the terms: -6 + -12r + 6r2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + -12r + 6r2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-1 + -2r + r2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-1 + -2r + r2)' equal to zero and attempt to solve: Simplifying -1 + -2r + r2 = 0 Solving -1 + -2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -2r + 1 + r2 = 0 + 1 Reorder the terms: -1 + 1 + -2r + r2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2r + r2 = 0 + 1 -2r + r2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2r + r2 = 1 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = 1 + 1 Reorder the terms: 1 + -2r + r2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2r + r2 = 2 Factor a perfect square on the left side: (r + -1)(r + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (r + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
r + -1 = 1.414213562 Simplifying r + -1 = 1.414213562 Reorder the terms: -1 + r = 1.414213562 Solving -1 + r = 1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + r = 1.414213562 + 1 r = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 r = 2.414213562 Simplifying r = 2.414213562Subproblem 2
r + -1 = -1.414213562 Simplifying r + -1 = -1.414213562 Reorder the terms: -1 + r = -1.414213562 Solving -1 + r = -1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + r = -1.414213562 + 1 r = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 r = -0.414213562 Simplifying r = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.414213562, -0.414213562}Solution
r = {2.414213562, -0.414213562}
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