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Simplifying (r + 4)(r + 2) = 12 Reorder the terms: (4 + r)(r + 2) = 12 Reorder the terms: (4 + r)(2 + r) = 12 Multiply (4 + r) * (2 + r) (4(2 + r) + r(2 + r)) = 12 ((2 * 4 + r * 4) + r(2 + r)) = 12 ((8 + 4r) + r(2 + r)) = 12 (8 + 4r + (2 * r + r * r)) = 12 (8 + 4r + (2r + r2)) = 12 Combine like terms: 4r + 2r = 6r (8 + 6r + r2) = 12 Solving 8 + 6r + r2 = 12 Solving for variable 'r'. Reorder the terms: 8 + -12 + 6r + r2 = 12 + -12 Combine like terms: 8 + -12 = -4 -4 + 6r + r2 = 12 + -12 Combine like terms: 12 + -12 = 0 -4 + 6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 6r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + 6r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 6r + r2 = 0 + 4 6r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 6r + r2 = 4 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 = 4 + 9 Reorder the terms: 9 + 6r + r2 = 4 + 9 Combine like terms: 4 + 9 = 13 9 + 6r + r2 = 13 Factor a perfect square on the left side: (r + 3)(r + 3) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (r + 3) equal to 3.605551275 and -3.605551275.Subproblem 1
r + 3 = 3.605551275 Simplifying r + 3 = 3.605551275 Reorder the terms: 3 + r = 3.605551275 Solving 3 + r = 3.605551275 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 = 3.605551275 + -3 Combine like terms: 3 + -3 = 0 0 + r = 3.605551275 + -3 r = 3.605551275 + -3 Combine like terms: 3.605551275 + -3 = 0.605551275 r = 0.605551275 Simplifying r = 0.605551275Subproblem 2
r + 3 = -3.605551275 Simplifying r + 3 = -3.605551275 Reorder the terms: 3 + r = -3.605551275 Solving 3 + r = -3.605551275 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 = -3.605551275 + -3 Combine like terms: 3 + -3 = 0 0 + r = -3.605551275 + -3 r = -3.605551275 + -3 Combine like terms: -3.605551275 + -3 = -6.605551275 r = -6.605551275 Simplifying r = -6.605551275Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.605551275, -6.605551275}
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