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Simplifying (r + -3)(r + 7) = -5 Reorder the terms: (-3 + r)(r + 7) = -5 Reorder the terms: (-3 + r)(7 + r) = -5 Multiply (-3 + r) * (7 + r) (-3(7 + r) + r(7 + r)) = -5 ((7 * -3 + r * -3) + r(7 + r)) = -5 ((-21 + -3r) + r(7 + r)) = -5 (-21 + -3r + (7 * r + r * r)) = -5 (-21 + -3r + (7r + r2)) = -5 Combine like terms: -3r + 7r = 4r (-21 + 4r + r2) = -5 Solving -21 + 4r + r2 = -5 Solving for variable 'r'. Reorder the terms: -21 + 5 + 4r + r2 = -5 + 5 Combine like terms: -21 + 5 = -16 -16 + 4r + r2 = -5 + 5 Combine like terms: -5 + 5 = 0 -16 + 4r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '16' to each side of the equation. -16 + 4r + 16 + r2 = 0 + 16 Reorder the terms: -16 + 16 + 4r + r2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 4r + r2 = 0 + 16 4r + r2 = 0 + 16 Combine like terms: 0 + 16 = 16 4r + r2 = 16 The r term is 4r. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4r + 4 + r2 = 16 + 4 Reorder the terms: 4 + 4r + r2 = 16 + 4 Combine like terms: 16 + 4 = 20 4 + 4r + r2 = 20 Factor a perfect square on the left side: (r + 2)(r + 2) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (r + 2) equal to 4.472135955 and -4.472135955.Subproblem 1
r + 2 = 4.472135955 Simplifying r + 2 = 4.472135955 Reorder the terms: 2 + r = 4.472135955 Solving 2 + r = 4.472135955 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = 4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + r = 4.472135955 + -2 r = 4.472135955 + -2 Combine like terms: 4.472135955 + -2 = 2.472135955 r = 2.472135955 Simplifying r = 2.472135955Subproblem 2
r + 2 = -4.472135955 Simplifying r + 2 = -4.472135955 Reorder the terms: 2 + r = -4.472135955 Solving 2 + r = -4.472135955 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = -4.472135955 + -2 Combine like terms: 2 + -2 = 0 0 + r = -4.472135955 + -2 r = -4.472135955 + -2 Combine like terms: -4.472135955 + -2 = -6.472135955 r = -6.472135955 Simplifying r = -6.472135955Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.472135955, -6.472135955}
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