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Simplifying -2r2 + 2r + 2 = 0 Reorder the terms: 2 + 2r + -2r2 = 0 Solving 2 + 2r + -2r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(1 + r + -1r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + r + -1r2)' equal to zero and attempt to solve: Simplifying 1 + r + -1r2 = 0 Solving 1 + r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -1r + r2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1r + 1 + r2 = 0 + 1 Reorder the terms: -1 + 1 + -1r + r2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1r + r2 = 0 + 1 -1r + r2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1r + r2 = 1 The r term is r. 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 = 1 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1r + r2 = 1.25 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (r + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
r + 0.5 = 1.118033989 Simplifying r + 0.5 = 1.118033989 Reorder the terms: 0.5 + r = 1.118033989 Solving 0.5 + r = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.118033989 + -0.5 r = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 r = 0.618033989 Simplifying r = 0.618033989Subproblem 2
r + 0.5 = -1.118033989 Simplifying r + 0.5 = -1.118033989 Reorder the terms: 0.5 + r = -1.118033989 Solving 0.5 + r = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.118033989 + -0.5 r = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 r = -1.618033989 Simplifying r = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.618033989, -1.618033989}Solution
r = {0.618033989, -1.618033989}
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