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Simplifying r2 + 5r + 5 = 0 Reorder the terms: 5 + 5r + r2 = 0 Solving 5 + 5r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 5r + -5 + r2 = 0 + -5 Reorder the terms: 5 + -5 + 5r + r2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 5r + r2 = 0 + -5 5r + r2 = 0 + -5 Combine like terms: 0 + -5 = -5 5r + r2 = -5 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = -5 + 6.25 Reorder the terms: 6.25 + 5r + r2 = -5 + 6.25 Combine like terms: -5 + 6.25 = 1.25 6.25 + 5r + r2 = 1.25 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (r + 2.5) equal to 1.118033989 and -1.118033989.Subproblem 1
r + 2.5 = 1.118033989 Simplifying r + 2.5 = 1.118033989 Reorder the terms: 2.5 + r = 1.118033989 Solving 2.5 + r = 1.118033989 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 1.118033989 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 1.118033989 + -2.5 r = 1.118033989 + -2.5 Combine like terms: 1.118033989 + -2.5 = -1.381966011 r = -1.381966011 Simplifying r = -1.381966011Subproblem 2
r + 2.5 = -1.118033989 Simplifying r + 2.5 = -1.118033989 Reorder the terms: 2.5 + r = -1.118033989 Solving 2.5 + r = -1.118033989 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -1.118033989 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -1.118033989 + -2.5 r = -1.118033989 + -2.5 Combine like terms: -1.118033989 + -2.5 = -3.618033989 r = -3.618033989 Simplifying r = -3.618033989Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.381966011, -3.618033989}
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