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Simplifying r2 + 2r + -19 = 0 Reorder the terms: -19 + 2r + r2 = 0 Solving -19 + 2r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '19' to each side of the equation. -19 + 2r + 19 + r2 = 0 + 19 Reorder the terms: -19 + 19 + 2r + r2 = 0 + 19 Combine like terms: -19 + 19 = 0 0 + 2r + r2 = 0 + 19 2r + r2 = 0 + 19 Combine like terms: 0 + 19 = 19 2r + r2 = 19 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 = 19 + 1 Reorder the terms: 1 + 2r + r2 = 19 + 1 Combine like terms: 19 + 1 = 20 1 + 2r + r2 = 20 Factor a perfect square on the left side: (r + 1)(r + 1) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (r + 1) equal to 4.472135955 and -4.472135955.Subproblem 1
r + 1 = 4.472135955 Simplifying r + 1 = 4.472135955 Reorder the terms: 1 + r = 4.472135955 Solving 1 + r = 4.472135955 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 = 4.472135955 + -1 Combine like terms: 1 + -1 = 0 0 + r = 4.472135955 + -1 r = 4.472135955 + -1 Combine like terms: 4.472135955 + -1 = 3.472135955 r = 3.472135955 Simplifying r = 3.472135955Subproblem 2
r + 1 = -4.472135955 Simplifying r + 1 = -4.472135955 Reorder the terms: 1 + r = -4.472135955 Solving 1 + r = -4.472135955 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 = -4.472135955 + -1 Combine like terms: 1 + -1 = 0 0 + r = -4.472135955 + -1 r = -4.472135955 + -1 Combine like terms: -4.472135955 + -1 = -5.472135955 r = -5.472135955 Simplifying r = -5.472135955Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.472135955, -5.472135955}
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