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Simplifying (r + -3)(r + 13) = -19 Reorder the terms: (-3 + r)(r + 13) = -19 Reorder the terms: (-3 + r)(13 + r) = -19 Multiply (-3 + r) * (13 + r) (-3(13 + r) + r(13 + r)) = -19 ((13 * -3 + r * -3) + r(13 + r)) = -19 ((-39 + -3r) + r(13 + r)) = -19 (-39 + -3r + (13 * r + r * r)) = -19 (-39 + -3r + (13r + r2)) = -19 Combine like terms: -3r + 13r = 10r (-39 + 10r + r2) = -19 Solving -39 + 10r + r2 = -19 Solving for variable 'r'. Reorder the terms: -39 + 19 + 10r + r2 = -19 + 19 Combine like terms: -39 + 19 = -20 -20 + 10r + r2 = -19 + 19 Combine like terms: -19 + 19 = 0 -20 + 10r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '20' to each side of the equation. -20 + 10r + 20 + r2 = 0 + 20 Reorder the terms: -20 + 20 + 10r + r2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 10r + r2 = 0 + 20 10r + r2 = 0 + 20 Combine like terms: 0 + 20 = 20 10r + r2 = 20 The r term is 10r. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10r + 25 + r2 = 20 + 25 Reorder the terms: 25 + 10r + r2 = 20 + 25 Combine like terms: 20 + 25 = 45 25 + 10r + r2 = 45 Factor a perfect square on the left side: (r + 5)(r + 5) = 45 Calculate the square root of the right side: 6.708203933 Break this problem into two subproblems by setting (r + 5) equal to 6.708203933 and -6.708203933.Subproblem 1
r + 5 = 6.708203933 Simplifying r + 5 = 6.708203933 Reorder the terms: 5 + r = 6.708203933 Solving 5 + r = 6.708203933 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = 6.708203933 + -5 Combine like terms: 5 + -5 = 0 0 + r = 6.708203933 + -5 r = 6.708203933 + -5 Combine like terms: 6.708203933 + -5 = 1.708203933 r = 1.708203933 Simplifying r = 1.708203933Subproblem 2
r + 5 = -6.708203933 Simplifying r + 5 = -6.708203933 Reorder the terms: 5 + r = -6.708203933 Solving 5 + r = -6.708203933 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = -6.708203933 + -5 Combine like terms: 5 + -5 = 0 0 + r = -6.708203933 + -5 r = -6.708203933 + -5 Combine like terms: -6.708203933 + -5 = -11.708203933 r = -11.708203933 Simplifying r = -11.708203933Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.708203933, -11.708203933}
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