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Simplifying -2r + -1r + 4 = 3r2 Reorder the terms: 4 + -2r + -1r = 3r2 Combine like terms: -2r + -1r = -3r 4 + -3r = 3r2 Solving 4 + -3r = 3r2 Solving for variable 'r'. Combine like terms: 3r2 + -3r2 = 0 4 + -3r + -3r2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.333333333 + r + r2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + r + 1.333333333 + r2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + r + r2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + r + r2 = 0 + 1.333333333 r + r2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 r + r2 = 1.333333333 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 1.333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 1.333333333 + 0.25 Combine like terms: 1.333333333 + 0.25 = 1.583333333 1.25 + r2 = 1.583333333 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (r + 0.5) equal to 1.258305739 and -1.258305739.Subproblem 1
r + 0.5 = 1.258305739 Simplifying r + 0.5 = 1.258305739 Reorder the terms: 0.5 + r = 1.258305739 Solving 0.5 + r = 1.258305739 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.258305739 + -0.5 r = 1.258305739 + -0.5 Combine like terms: 1.258305739 + -0.5 = 0.758305739 r = 0.758305739 Simplifying r = 0.758305739Subproblem 2
r + 0.5 = -1.258305739 Simplifying r + 0.5 = -1.258305739 Reorder the terms: 0.5 + r = -1.258305739 Solving 0.5 + r = -1.258305739 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.258305739 + -0.5 r = -1.258305739 + -0.5 Combine like terms: -1.258305739 + -0.5 = -1.758305739 r = -1.758305739 Simplifying r = -1.758305739Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.758305739, -1.758305739}
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