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Simplifying (r + -2)(r + 7) = 9 Reorder the terms: (-2 + r)(r + 7) = 9 Reorder the terms: (-2 + r)(7 + r) = 9 Multiply (-2 + r) * (7 + r) (-2(7 + r) + r(7 + r)) = 9 ((7 * -2 + r * -2) + r(7 + r)) = 9 ((-14 + -2r) + r(7 + r)) = 9 (-14 + -2r + (7 * r + r * r)) = 9 (-14 + -2r + (7r + r2)) = 9 Combine like terms: -2r + 7r = 5r (-14 + 5r + r2) = 9 Solving -14 + 5r + r2 = 9 Solving for variable 'r'. Reorder the terms: -14 + -9 + 5r + r2 = 9 + -9 Combine like terms: -14 + -9 = -23 -23 + 5r + r2 = 9 + -9 Combine like terms: 9 + -9 = 0 -23 + 5r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 5r + 23 + r2 = 0 + 23 Reorder the terms: -23 + 23 + 5r + r2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 5r + r2 = 0 + 23 5r + r2 = 0 + 23 Combine like terms: 0 + 23 = 23 5r + r2 = 23 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 = 23 + 6.25 Reorder the terms: 6.25 + 5r + r2 = 23 + 6.25 Combine like terms: 23 + 6.25 = 29.25 6.25 + 5r + r2 = 29.25 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 29.25 Calculate the square root of the right side: 5.408326913 Break this problem into two subproblems by setting (r + 2.5) equal to 5.408326913 and -5.408326913.Subproblem 1
r + 2.5 = 5.408326913 Simplifying r + 2.5 = 5.408326913 Reorder the terms: 2.5 + r = 5.408326913 Solving 2.5 + r = 5.408326913 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 = 5.408326913 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 5.408326913 + -2.5 r = 5.408326913 + -2.5 Combine like terms: 5.408326913 + -2.5 = 2.908326913 r = 2.908326913 Simplifying r = 2.908326913Subproblem 2
r + 2.5 = -5.408326913 Simplifying r + 2.5 = -5.408326913 Reorder the terms: 2.5 + r = -5.408326913 Solving 2.5 + r = -5.408326913 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 = -5.408326913 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -5.408326913 + -2.5 r = -5.408326913 + -2.5 Combine like terms: -5.408326913 + -2.5 = -7.908326913 r = -7.908326913 Simplifying r = -7.908326913Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.908326913, -7.908326913}
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