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Simplifying (r + -4)(r + 10) = -22 Reorder the terms: (-4 + r)(r + 10) = -22 Reorder the terms: (-4 + r)(10 + r) = -22 Multiply (-4 + r) * (10 + r) (-4(10 + r) + r(10 + r)) = -22 ((10 * -4 + r * -4) + r(10 + r)) = -22 ((-40 + -4r) + r(10 + r)) = -22 (-40 + -4r + (10 * r + r * r)) = -22 (-40 + -4r + (10r + r2)) = -22 Combine like terms: -4r + 10r = 6r (-40 + 6r + r2) = -22 Solving -40 + 6r + r2 = -22 Solving for variable 'r'. Reorder the terms: -40 + 22 + 6r + r2 = -22 + 22 Combine like terms: -40 + 22 = -18 -18 + 6r + r2 = -22 + 22 Combine like terms: -22 + 22 = 0 -18 + 6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '18' to each side of the equation. -18 + 6r + 18 + r2 = 0 + 18 Reorder the terms: -18 + 18 + 6r + r2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 6r + r2 = 0 + 18 6r + r2 = 0 + 18 Combine like terms: 0 + 18 = 18 6r + r2 = 18 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 = 18 + 9 Reorder the terms: 9 + 6r + r2 = 18 + 9 Combine like terms: 18 + 9 = 27 9 + 6r + r2 = 27 Factor a perfect square on the left side: (r + 3)(r + 3) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (r + 3) equal to 5.196152423 and -5.196152423.Subproblem 1
r + 3 = 5.196152423 Simplifying r + 3 = 5.196152423 Reorder the terms: 3 + r = 5.196152423 Solving 3 + r = 5.196152423 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 = 5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + r = 5.196152423 + -3 r = 5.196152423 + -3 Combine like terms: 5.196152423 + -3 = 2.196152423 r = 2.196152423 Simplifying r = 2.196152423Subproblem 2
r + 3 = -5.196152423 Simplifying r + 3 = -5.196152423 Reorder the terms: 3 + r = -5.196152423 Solving 3 + r = -5.196152423 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 = -5.196152423 + -3 Combine like terms: 3 + -3 = 0 0 + r = -5.196152423 + -3 r = -5.196152423 + -3 Combine like terms: -5.196152423 + -3 = -8.196152423 r = -8.196152423 Simplifying r = -8.196152423Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.196152423, -8.196152423}
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