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Simplifying r2 + 12r + 31 = 0 Reorder the terms: 31 + 12r + r2 = 0 Solving 31 + 12r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '-31' to each side of the equation. 31 + 12r + -31 + r2 = 0 + -31 Reorder the terms: 31 + -31 + 12r + r2 = 0 + -31 Combine like terms: 31 + -31 = 0 0 + 12r + r2 = 0 + -31 12r + r2 = 0 + -31 Combine like terms: 0 + -31 = -31 12r + r2 = -31 The r term is 12r. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12r + 36 + r2 = -31 + 36 Reorder the terms: 36 + 12r + r2 = -31 + 36 Combine like terms: -31 + 36 = 5 36 + 12r + r2 = 5 Factor a perfect square on the left side: (r + 6)(r + 6) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (r + 6) equal to 2.236067978 and -2.236067978.Subproblem 1
r + 6 = 2.236067978 Simplifying r + 6 = 2.236067978 Reorder the terms: 6 + r = 2.236067978 Solving 6 + r = 2.236067978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = 2.236067978 + -6 Combine like terms: 6 + -6 = 0 0 + r = 2.236067978 + -6 r = 2.236067978 + -6 Combine like terms: 2.236067978 + -6 = -3.763932022 r = -3.763932022 Simplifying r = -3.763932022Subproblem 2
r + 6 = -2.236067978 Simplifying r + 6 = -2.236067978 Reorder the terms: 6 + r = -2.236067978 Solving 6 + r = -2.236067978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = -2.236067978 + -6 Combine like terms: 6 + -6 = 0 0 + r = -2.236067978 + -6 r = -2.236067978 + -6 Combine like terms: -2.236067978 + -6 = -8.236067978 r = -8.236067978 Simplifying r = -8.236067978Solution
The solution to the problem is based on the solutions from the subproblems. r = {-3.763932022, -8.236067978}
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