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Simplifying 2r2 + 15r + 23 = 0 Reorder the terms: 23 + 15r + 2r2 = 0 Solving 23 + 15r + 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'. 11.5 + 7.5r + r2 = 0 Move the constant term to the right: Add '-11.5' to each side of the equation. 11.5 + 7.5r + -11.5 + r2 = 0 + -11.5 Reorder the terms: 11.5 + -11.5 + 7.5r + r2 = 0 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + 7.5r + r2 = 0 + -11.5 7.5r + r2 = 0 + -11.5 Combine like terms: 0 + -11.5 = -11.5 7.5r + r2 = -11.5 The r term is 7.5r. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5r + 14.0625 + r2 = -11.5 + 14.0625 Reorder the terms: 14.0625 + 7.5r + r2 = -11.5 + 14.0625 Combine like terms: -11.5 + 14.0625 = 2.5625 14.0625 + 7.5r + r2 = 2.5625 Factor a perfect square on the left side: (r + 3.75)(r + 3.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (r + 3.75) equal to 1.600781059 and -1.600781059.Subproblem 1
r + 3.75 = 1.600781059 Simplifying r + 3.75 = 1.600781059 Reorder the terms: 3.75 + r = 1.600781059 Solving 3.75 + r = 1.600781059 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = 1.600781059 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = 1.600781059 + -3.75 r = 1.600781059 + -3.75 Combine like terms: 1.600781059 + -3.75 = -2.149218941 r = -2.149218941 Simplifying r = -2.149218941Subproblem 2
r + 3.75 = -1.600781059 Simplifying r + 3.75 = -1.600781059 Reorder the terms: 3.75 + r = -1.600781059 Solving 3.75 + r = -1.600781059 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = -1.600781059 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = -1.600781059 + -3.75 r = -1.600781059 + -3.75 Combine like terms: -1.600781059 + -3.75 = -5.350781059 r = -5.350781059 Simplifying r = -5.350781059Solution
The solution to the problem is based on the solutions from the subproblems. r = {-2.149218941, -5.350781059}
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