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Simplifying (r + -3)(r + 7) = 100 Reorder the terms: (-3 + r)(r + 7) = 100 Reorder the terms: (-3 + r)(7 + r) = 100 Multiply (-3 + r) * (7 + r) (-3(7 + r) + r(7 + r)) = 100 ((7 * -3 + r * -3) + r(7 + r)) = 100 ((-21 + -3r) + r(7 + r)) = 100 (-21 + -3r + (7 * r + r * r)) = 100 (-21 + -3r + (7r + r2)) = 100 Combine like terms: -3r + 7r = 4r (-21 + 4r + r2) = 100 Solving -21 + 4r + r2 = 100 Solving for variable 'r'. Reorder the terms: -21 + -100 + 4r + r2 = 100 + -100 Combine like terms: -21 + -100 = -121 -121 + 4r + r2 = 100 + -100 Combine like terms: 100 + -100 = 0 -121 + 4r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '121' to each side of the equation. -121 + 4r + 121 + r2 = 0 + 121 Reorder the terms: -121 + 121 + 4r + r2 = 0 + 121 Combine like terms: -121 + 121 = 0 0 + 4r + r2 = 0 + 121 4r + r2 = 0 + 121 Combine like terms: 0 + 121 = 121 4r + r2 = 121 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 = 121 + 4 Reorder the terms: 4 + 4r + r2 = 121 + 4 Combine like terms: 121 + 4 = 125 4 + 4r + r2 = 125 Factor a perfect square on the left side: (r + 2)(r + 2) = 125 Calculate the square root of the right side: 11.180339887 Break this problem into two subproblems by setting (r + 2) equal to 11.180339887 and -11.180339887.Subproblem 1
r + 2 = 11.180339887 Simplifying r + 2 = 11.180339887 Reorder the terms: 2 + r = 11.180339887 Solving 2 + r = 11.180339887 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 = 11.180339887 + -2 Combine like terms: 2 + -2 = 0 0 + r = 11.180339887 + -2 r = 11.180339887 + -2 Combine like terms: 11.180339887 + -2 = 9.180339887 r = 9.180339887 Simplifying r = 9.180339887Subproblem 2
r + 2 = -11.180339887 Simplifying r + 2 = -11.180339887 Reorder the terms: 2 + r = -11.180339887 Solving 2 + r = -11.180339887 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 = -11.180339887 + -2 Combine like terms: 2 + -2 = 0 0 + r = -11.180339887 + -2 r = -11.180339887 + -2 Combine like terms: -11.180339887 + -2 = -13.180339887 r = -13.180339887 Simplifying r = -13.180339887Solution
The solution to the problem is based on the solutions from the subproblems. r = {9.180339887, -13.180339887}
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