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Simplifying (r + -4)(r + 18) = -41 Reorder the terms: (-4 + r)(r + 18) = -41 Reorder the terms: (-4 + r)(18 + r) = -41 Multiply (-4 + r) * (18 + r) (-4(18 + r) + r(18 + r)) = -41 ((18 * -4 + r * -4) + r(18 + r)) = -41 ((-72 + -4r) + r(18 + r)) = -41 (-72 + -4r + (18 * r + r * r)) = -41 (-72 + -4r + (18r + r2)) = -41 Combine like terms: -4r + 18r = 14r (-72 + 14r + r2) = -41 Solving -72 + 14r + r2 = -41 Solving for variable 'r'. Reorder the terms: -72 + 41 + 14r + r2 = -41 + 41 Combine like terms: -72 + 41 = -31 -31 + 14r + r2 = -41 + 41 Combine like terms: -41 + 41 = 0 -31 + 14r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '31' to each side of the equation. -31 + 14r + 31 + r2 = 0 + 31 Reorder the terms: -31 + 31 + 14r + r2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 14r + r2 = 0 + 31 14r + r2 = 0 + 31 Combine like terms: 0 + 31 = 31 14r + r2 = 31 The r term is 14r. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14r + 49 + r2 = 31 + 49 Reorder the terms: 49 + 14r + r2 = 31 + 49 Combine like terms: 31 + 49 = 80 49 + 14r + r2 = 80 Factor a perfect square on the left side: (r + 7)(r + 7) = 80 Calculate the square root of the right side: 8.94427191 Break this problem into two subproblems by setting (r + 7) equal to 8.94427191 and -8.94427191.Subproblem 1
r + 7 = 8.94427191 Simplifying r + 7 = 8.94427191 Reorder the terms: 7 + r = 8.94427191 Solving 7 + r = 8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + r = 8.94427191 + -7 Combine like terms: 7 + -7 = 0 0 + r = 8.94427191 + -7 r = 8.94427191 + -7 Combine like terms: 8.94427191 + -7 = 1.94427191 r = 1.94427191 Simplifying r = 1.94427191Subproblem 2
r + 7 = -8.94427191 Simplifying r + 7 = -8.94427191 Reorder the terms: 7 + r = -8.94427191 Solving 7 + r = -8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + r = -8.94427191 + -7 Combine like terms: 7 + -7 = 0 0 + r = -8.94427191 + -7 r = -8.94427191 + -7 Combine like terms: -8.94427191 + -7 = -15.94427191 r = -15.94427191 Simplifying r = -15.94427191Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.94427191, -15.94427191}
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