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Simplifying 36 = (2r + 3)(r + -3) Reorder the terms: 36 = (3 + 2r)(r + -3) Reorder the terms: 36 = (3 + 2r)(-3 + r) Multiply (3 + 2r) * (-3 + r) 36 = (3(-3 + r) + 2r * (-3 + r)) 36 = ((-3 * 3 + r * 3) + 2r * (-3 + r)) 36 = ((-9 + 3r) + 2r * (-3 + r)) 36 = (-9 + 3r + (-3 * 2r + r * 2r)) 36 = (-9 + 3r + (-6r + 2r2)) Combine like terms: 3r + -6r = -3r 36 = (-9 + -3r + 2r2) Solving 36 = -9 + -3r + 2r2 Solving for variable 'r'. Combine like terms: 36 + 9 = 45 45 + 3r + -2r2 = -9 + -3r + 2r2 + 9 + 3r + -2r2 Reorder the terms: 45 + 3r + -2r2 = -9 + 9 + -3r + 3r + 2r2 + -2r2 Combine like terms: -9 + 9 = 0 45 + 3r + -2r2 = 0 + -3r + 3r + 2r2 + -2r2 45 + 3r + -2r2 = -3r + 3r + 2r2 + -2r2 Combine like terms: -3r + 3r = 0 45 + 3r + -2r2 = 0 + 2r2 + -2r2 45 + 3r + -2r2 = 2r2 + -2r2 Combine like terms: 2r2 + -2r2 = 0 45 + 3r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -22.5 + -1.5r + r2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + -1.5r + 22.5 + r2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + -1.5r + r2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + -1.5r + r2 = 0 + 22.5 -1.5r + r2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 -1.5r + r2 = 22.5 The r term is -1.5r. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5r + 0.5625 + r2 = 22.5 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = 22.5 + 0.5625 Combine like terms: 22.5 + 0.5625 = 23.0625 0.5625 + -1.5r + r2 = 23.0625 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 23.0625 Calculate the square root of the right side: 4.802343178 Break this problem into two subproblems by setting (r + -0.75) equal to 4.802343178 and -4.802343178.Subproblem 1
r + -0.75 = 4.802343178 Simplifying r + -0.75 = 4.802343178 Reorder the terms: -0.75 + r = 4.802343178 Solving -0.75 + r = 4.802343178 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 4.802343178 + 0.75 r = 4.802343178 + 0.75 Combine like terms: 4.802343178 + 0.75 = 5.552343178 r = 5.552343178 Simplifying r = 5.552343178Subproblem 2
r + -0.75 = -4.802343178 Simplifying r + -0.75 = -4.802343178 Reorder the terms: -0.75 + r = -4.802343178 Solving -0.75 + r = -4.802343178 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -4.802343178 + 0.75 r = -4.802343178 + 0.75 Combine like terms: -4.802343178 + 0.75 = -4.052343178 r = -4.052343178 Simplifying r = -4.052343178Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.552343178, -4.052343178}
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