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Simplifying (r + -4)(r + 10) = 31 Reorder the terms: (-4 + r)(r + 10) = 31 Reorder the terms: (-4 + r)(10 + r) = 31 Multiply (-4 + r) * (10 + r) (-4(10 + r) + r(10 + r)) = 31 ((10 * -4 + r * -4) + r(10 + r)) = 31 ((-40 + -4r) + r(10 + r)) = 31 (-40 + -4r + (10 * r + r * r)) = 31 (-40 + -4r + (10r + r2)) = 31 Combine like terms: -4r + 10r = 6r (-40 + 6r + r2) = 31 Solving -40 + 6r + r2 = 31 Solving for variable 'r'. Reorder the terms: -40 + -31 + 6r + r2 = 31 + -31 Combine like terms: -40 + -31 = -71 -71 + 6r + r2 = 31 + -31 Combine like terms: 31 + -31 = 0 -71 + 6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '71' to each side of the equation. -71 + 6r + 71 + r2 = 0 + 71 Reorder the terms: -71 + 71 + 6r + r2 = 0 + 71 Combine like terms: -71 + 71 = 0 0 + 6r + r2 = 0 + 71 6r + r2 = 0 + 71 Combine like terms: 0 + 71 = 71 6r + r2 = 71 The r term is 6r. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6r + 9 + r2 = 71 + 9 Reorder the terms: 9 + 6r + r2 = 71 + 9 Combine like terms: 71 + 9 = 80 9 + 6r + r2 = 80 Factor a perfect square on the left side: (r + 3)(r + 3) = 80 Calculate the square root of the right side: 8.94427191 Break this problem into two subproblems by setting (r + 3) equal to 8.94427191 and -8.94427191.Subproblem 1
r + 3 = 8.94427191 Simplifying r + 3 = 8.94427191 Reorder the terms: 3 + r = 8.94427191 Solving 3 + r = 8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 8.94427191 + -3 Combine like terms: 3 + -3 = 0 0 + r = 8.94427191 + -3 r = 8.94427191 + -3 Combine like terms: 8.94427191 + -3 = 5.94427191 r = 5.94427191 Simplifying r = 5.94427191Subproblem 2
r + 3 = -8.94427191 Simplifying r + 3 = -8.94427191 Reorder the terms: 3 + r = -8.94427191 Solving 3 + r = -8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -8.94427191 + -3 Combine like terms: 3 + -3 = 0 0 + r = -8.94427191 + -3 r = -8.94427191 + -3 Combine like terms: -8.94427191 + -3 = -11.94427191 r = -11.94427191 Simplifying r = -11.94427191Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.94427191, -11.94427191}
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