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Simplifying r2 + 2r = 47 Reorder the terms: 2r + r2 = 47 Solving 2r + r2 = 47 Solving for variable 'r'. Reorder the terms: -47 + 2r + r2 = 47 + -47 Combine like terms: 47 + -47 = 0 -47 + 2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '47' to each side of the equation. -47 + 2r + 47 + r2 = 0 + 47 Reorder the terms: -47 + 47 + 2r + r2 = 0 + 47 Combine like terms: -47 + 47 = 0 0 + 2r + r2 = 0 + 47 2r + r2 = 0 + 47 Combine like terms: 0 + 47 = 47 2r + r2 = 47 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 = 47 + 1 Reorder the terms: 1 + 2r + r2 = 47 + 1 Combine like terms: 47 + 1 = 48 1 + 2r + r2 = 48 Factor a perfect square on the left side: (r + 1)(r + 1) = 48 Calculate the square root of the right side: 6.92820323 Break this problem into two subproblems by setting (r + 1) equal to 6.92820323 and -6.92820323.Subproblem 1
r + 1 = 6.92820323 Simplifying r + 1 = 6.92820323 Reorder the terms: 1 + r = 6.92820323 Solving 1 + r = 6.92820323 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 = 6.92820323 + -1 Combine like terms: 1 + -1 = 0 0 + r = 6.92820323 + -1 r = 6.92820323 + -1 Combine like terms: 6.92820323 + -1 = 5.92820323 r = 5.92820323 Simplifying r = 5.92820323Subproblem 2
r + 1 = -6.92820323 Simplifying r + 1 = -6.92820323 Reorder the terms: 1 + r = -6.92820323 Solving 1 + r = -6.92820323 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 = -6.92820323 + -1 Combine like terms: 1 + -1 = 0 0 + r = -6.92820323 + -1 r = -6.92820323 + -1 Combine like terms: -6.92820323 + -1 = -7.92820323 r = -7.92820323 Simplifying r = -7.92820323Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.92820323, -7.92820323}
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