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Simplifying 2r2 + 18r + 33 = 0 Reorder the terms: 33 + 18r + 2r2 = 0 Solving 33 + 18r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 16.5 + 9r + r2 = 0 Move the constant term to the right: Add '-16.5' to each side of the equation. 16.5 + 9r + -16.5 + r2 = 0 + -16.5 Reorder the terms: 16.5 + -16.5 + 9r + r2 = 0 + -16.5 Combine like terms: 16.5 + -16.5 = 0.0 0.0 + 9r + r2 = 0 + -16.5 9r + r2 = 0 + -16.5 Combine like terms: 0 + -16.5 = -16.5 9r + r2 = -16.5 The r term is 9r. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9r + 20.25 + r2 = -16.5 + 20.25 Reorder the terms: 20.25 + 9r + r2 = -16.5 + 20.25 Combine like terms: -16.5 + 20.25 = 3.75 20.25 + 9r + r2 = 3.75 Factor a perfect square on the left side: (r + 4.5)(r + 4.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (r + 4.5) equal to 1.936491673 and -1.936491673.Subproblem 1
r + 4.5 = 1.936491673 Simplifying r + 4.5 = 1.936491673 Reorder the terms: 4.5 + r = 1.936491673 Solving 4.5 + r = 1.936491673 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + r = 1.936491673 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + r = 1.936491673 + -4.5 r = 1.936491673 + -4.5 Combine like terms: 1.936491673 + -4.5 = -2.563508327 r = -2.563508327 Simplifying r = -2.563508327Subproblem 2
r + 4.5 = -1.936491673 Simplifying r + 4.5 = -1.936491673 Reorder the terms: 4.5 + r = -1.936491673 Solving 4.5 + r = -1.936491673 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + r = -1.936491673 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + r = -1.936491673 + -4.5 r = -1.936491673 + -4.5 Combine like terms: -1.936491673 + -4.5 = -6.436491673 r = -6.436491673 Simplifying r = -6.436491673Solution
The solution to the problem is based on the solutions from the subproblems. r = {-2.563508327, -6.436491673}
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