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Simplifying (r + -3)(r + 7) = 23 Reorder the terms: (-3 + r)(r + 7) = 23 Reorder the terms: (-3 + r)(7 + r) = 23 Multiply (-3 + r) * (7 + r) (-3(7 + r) + r(7 + r)) = 23 ((7 * -3 + r * -3) + r(7 + r)) = 23 ((-21 + -3r) + r(7 + r)) = 23 (-21 + -3r + (7 * r + r * r)) = 23 (-21 + -3r + (7r + r2)) = 23 Combine like terms: -3r + 7r = 4r (-21 + 4r + r2) = 23 Solving -21 + 4r + r2 = 23 Solving for variable 'r'. Reorder the terms: -21 + -23 + 4r + r2 = 23 + -23 Combine like terms: -21 + -23 = -44 -44 + 4r + r2 = 23 + -23 Combine like terms: 23 + -23 = 0 -44 + 4r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '44' to each side of the equation. -44 + 4r + 44 + r2 = 0 + 44 Reorder the terms: -44 + 44 + 4r + r2 = 0 + 44 Combine like terms: -44 + 44 = 0 0 + 4r + r2 = 0 + 44 4r + r2 = 0 + 44 Combine like terms: 0 + 44 = 44 4r + r2 = 44 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 = 44 + 4 Reorder the terms: 4 + 4r + r2 = 44 + 4 Combine like terms: 44 + 4 = 48 4 + 4r + r2 = 48 Factor a perfect square on the left side: (r + 2)(r + 2) = 48 Calculate the square root of the right side: 6.92820323 Break this problem into two subproblems by setting (r + 2) equal to 6.92820323 and -6.92820323.Subproblem 1
r + 2 = 6.92820323 Simplifying r + 2 = 6.92820323 Reorder the terms: 2 + r = 6.92820323 Solving 2 + r = 6.92820323 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 = 6.92820323 + -2 Combine like terms: 2 + -2 = 0 0 + r = 6.92820323 + -2 r = 6.92820323 + -2 Combine like terms: 6.92820323 + -2 = 4.92820323 r = 4.92820323 Simplifying r = 4.92820323Subproblem 2
r + 2 = -6.92820323 Simplifying r + 2 = -6.92820323 Reorder the terms: 2 + r = -6.92820323 Solving 2 + r = -6.92820323 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 = -6.92820323 + -2 Combine like terms: 2 + -2 = 0 0 + r = -6.92820323 + -2 r = -6.92820323 + -2 Combine like terms: -6.92820323 + -2 = -8.92820323 r = -8.92820323 Simplifying r = -8.92820323Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.92820323, -8.92820323}
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