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Simplifying (r + -7)(r + 5) = 20 Reorder the terms: (-7 + r)(r + 5) = 20 Reorder the terms: (-7 + r)(5 + r) = 20 Multiply (-7 + r) * (5 + r) (-7(5 + r) + r(5 + r)) = 20 ((5 * -7 + r * -7) + r(5 + r)) = 20 ((-35 + -7r) + r(5 + r)) = 20 (-35 + -7r + (5 * r + r * r)) = 20 (-35 + -7r + (5r + r2)) = 20 Combine like terms: -7r + 5r = -2r (-35 + -2r + r2) = 20 Solving -35 + -2r + r2 = 20 Solving for variable 'r'. Reorder the terms: -35 + -20 + -2r + r2 = 20 + -20 Combine like terms: -35 + -20 = -55 -55 + -2r + r2 = 20 + -20 Combine like terms: 20 + -20 = 0 -55 + -2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '55' to each side of the equation. -55 + -2r + 55 + r2 = 0 + 55 Reorder the terms: -55 + 55 + -2r + r2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + -2r + r2 = 0 + 55 -2r + r2 = 0 + 55 Combine like terms: 0 + 55 = 55 -2r + r2 = 55 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 = 55 + 1 Reorder the terms: 1 + -2r + r2 = 55 + 1 Combine like terms: 55 + 1 = 56 1 + -2r + r2 = 56 Factor a perfect square on the left side: (r + -1)(r + -1) = 56 Calculate the square root of the right side: 7.483314774 Break this problem into two subproblems by setting (r + -1) equal to 7.483314774 and -7.483314774.Subproblem 1
r + -1 = 7.483314774 Simplifying r + -1 = 7.483314774 Reorder the terms: -1 + r = 7.483314774 Solving -1 + r = 7.483314774 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 = 7.483314774 + 1 Combine like terms: -1 + 1 = 0 0 + r = 7.483314774 + 1 r = 7.483314774 + 1 Combine like terms: 7.483314774 + 1 = 8.483314774 r = 8.483314774 Simplifying r = 8.483314774Subproblem 2
r + -1 = -7.483314774 Simplifying r + -1 = -7.483314774 Reorder the terms: -1 + r = -7.483314774 Solving -1 + r = -7.483314774 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 = -7.483314774 + 1 Combine like terms: -1 + 1 = 0 0 + r = -7.483314774 + 1 r = -7.483314774 + 1 Combine like terms: -7.483314774 + 1 = -6.483314774 r = -6.483314774 Simplifying r = -6.483314774Solution
The solution to the problem is based on the solutions from the subproblems. r = {8.483314774, -6.483314774}
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