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Simplifying 98r2 + 196r + -23 = 0 Reorder the terms: -23 + 196r + 98r2 = 0 Solving -23 + 196r + 98r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 98 the coefficient of the squared term: Divide each side by '98'. -0.2346938776 + 2r + r2 = 0 Move the constant term to the right: Add '0.2346938776' to each side of the equation. -0.2346938776 + 2r + 0.2346938776 + r2 = 0 + 0.2346938776 Reorder the terms: -0.2346938776 + 0.2346938776 + 2r + r2 = 0 + 0.2346938776 Combine like terms: -0.2346938776 + 0.2346938776 = 0.0000000000 0.0000000000 + 2r + r2 = 0 + 0.2346938776 2r + r2 = 0 + 0.2346938776 Combine like terms: 0 + 0.2346938776 = 0.2346938776 2r + r2 = 0.2346938776 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 0.2346938776 + 1 Reorder the terms: 1 + 2r + r2 = 0.2346938776 + 1 Combine like terms: 0.2346938776 + 1 = 1.2346938776 1 + 2r + r2 = 1.2346938776 Factor a perfect square on the left side: (r + 1)(r + 1) = 1.2346938776 Calculate the square root of the right side: 1.111167799 Break this problem into two subproblems by setting (r + 1) equal to 1.111167799 and -1.111167799.Subproblem 1
r + 1 = 1.111167799 Simplifying r + 1 = 1.111167799 Reorder the terms: 1 + r = 1.111167799 Solving 1 + r = 1.111167799 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 1.111167799 + -1 Combine like terms: 1 + -1 = 0 0 + r = 1.111167799 + -1 r = 1.111167799 + -1 Combine like terms: 1.111167799 + -1 = 0.111167799 r = 0.111167799 Simplifying r = 0.111167799Subproblem 2
r + 1 = -1.111167799 Simplifying r + 1 = -1.111167799 Reorder the terms: 1 + r = -1.111167799 Solving 1 + r = -1.111167799 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -1.111167799 + -1 Combine like terms: 1 + -1 = 0 0 + r = -1.111167799 + -1 r = -1.111167799 + -1 Combine like terms: -1.111167799 + -1 = -2.111167799 r = -2.111167799 Simplifying r = -2.111167799Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.111167799, -2.111167799}
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