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Simplifying -2r(r + 3) = -23 Reorder the terms: -2r(3 + r) = -23 (3 * -2r + r * -2r) = -23 (-6r + -2r2) = -23 Solving -6r + -2r2 = -23 Solving for variable 'r'. Reorder the terms: 23 + -6r + -2r2 = -23 + 23 Combine like terms: -23 + 23 = 0 23 + -6r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11.5 + 3r + r2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 3r + 11.5 + r2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 3r + r2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 3r + r2 = 0 + 11.5 3r + r2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 3r + r2 = 11.5 The r term is 3r. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3r + 2.25 + r2 = 11.5 + 2.25 Reorder the terms: 2.25 + 3r + r2 = 11.5 + 2.25 Combine like terms: 11.5 + 2.25 = 13.75 2.25 + 3r + r2 = 13.75 Factor a perfect square on the left side: (r + 1.5)(r + 1.5) = 13.75 Calculate the square root of the right side: 3.708099244 Break this problem into two subproblems by setting (r + 1.5) equal to 3.708099244 and -3.708099244.Subproblem 1
r + 1.5 = 3.708099244 Simplifying r + 1.5 = 3.708099244 Reorder the terms: 1.5 + r = 3.708099244 Solving 1.5 + r = 3.708099244 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 3.708099244 + -1.5 r = 3.708099244 + -1.5 Combine like terms: 3.708099244 + -1.5 = 2.208099244 r = 2.208099244 Simplifying r = 2.208099244Subproblem 2
r + 1.5 = -3.708099244 Simplifying r + 1.5 = -3.708099244 Reorder the terms: 1.5 + r = -3.708099244 Solving 1.5 + r = -3.708099244 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -3.708099244 + -1.5 r = -3.708099244 + -1.5 Combine like terms: -3.708099244 + -1.5 = -5.208099244 r = -5.208099244 Simplifying r = -5.208099244Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.208099244, -5.208099244}
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