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Simplifying 2r + -7(9 + 3r) * r = 5 Reorder the terms for easier multiplication: 2r + -7r(9 + 3r) = 5 2r + (9 * -7r + 3r * -7r) = 5 2r + (-63r + -21r2) = 5 Combine like terms: 2r + -63r = -61r -61r + -21r2 = 5 Solving -61r + -21r2 = 5 Solving for variable 'r'. Reorder the terms: -5 + -61r + -21r2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -61r + -21r2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 61r + 21r2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 61r + 21r2)' equal to zero and attempt to solve: Simplifying 5 + 61r + 21r2 = 0 Solving 5 + 61r + 21r2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 0.2380952381 + 2.904761905r + r2 = 0 Move the constant term to the right: Add '-0.2380952381' to each side of the equation. 0.2380952381 + 2.904761905r + -0.2380952381 + r2 = 0 + -0.2380952381 Reorder the terms: 0.2380952381 + -0.2380952381 + 2.904761905r + r2 = 0 + -0.2380952381 Combine like terms: 0.2380952381 + -0.2380952381 = 0.0000000000 0.0000000000 + 2.904761905r + r2 = 0 + -0.2380952381 2.904761905r + r2 = 0 + -0.2380952381 Combine like terms: 0 + -0.2380952381 = -0.2380952381 2.904761905r + r2 = -0.2380952381 The r term is 2.904761905r. Take half its coefficient (1.452380953). Square it (2.109410433) and add it to both sides. Add '2.109410433' to each side of the equation. 2.904761905r + 2.109410433 + r2 = -0.2380952381 + 2.109410433 Reorder the terms: 2.109410433 + 2.904761905r + r2 = -0.2380952381 + 2.109410433 Combine like terms: -0.2380952381 + 2.109410433 = 1.8713151949 2.109410433 + 2.904761905r + r2 = 1.8713151949 Factor a perfect square on the left side: (r + 1.452380953)(r + 1.452380953) = 1.8713151949 Calculate the square root of the right side: 1.367960231 Break this problem into two subproblems by setting (r + 1.452380953) equal to 1.367960231 and -1.367960231.Subproblem 1
r + 1.452380953 = 1.367960231 Simplifying r + 1.452380953 = 1.367960231 Reorder the terms: 1.452380953 + r = 1.367960231 Solving 1.452380953 + r = 1.367960231 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.452380953' to each side of the equation. 1.452380953 + -1.452380953 + r = 1.367960231 + -1.452380953 Combine like terms: 1.452380953 + -1.452380953 = 0.000000000 0.000000000 + r = 1.367960231 + -1.452380953 r = 1.367960231 + -1.452380953 Combine like terms: 1.367960231 + -1.452380953 = -0.084420722 r = -0.084420722 Simplifying r = -0.084420722Subproblem 2
r + 1.452380953 = -1.367960231 Simplifying r + 1.452380953 = -1.367960231 Reorder the terms: 1.452380953 + r = -1.367960231 Solving 1.452380953 + r = -1.367960231 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.452380953' to each side of the equation. 1.452380953 + -1.452380953 + r = -1.367960231 + -1.452380953 Combine like terms: 1.452380953 + -1.452380953 = 0.000000000 0.000000000 + r = -1.367960231 + -1.452380953 r = -1.367960231 + -1.452380953 Combine like terms: -1.367960231 + -1.452380953 = -2.820341184 r = -2.820341184 Simplifying r = -2.820341184Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.084420722, -2.820341184}Solution
r = {-0.084420722, -2.820341184}
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