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Simplifying 6r(7r + -3) + 3r(10r + 9) = 1 Reorder the terms: 6r(-3 + 7r) + 3r(10r + 9) = 1 (-3 * 6r + 7r * 6r) + 3r(10r + 9) = 1 (-18r + 42r2) + 3r(10r + 9) = 1 Reorder the terms: -18r + 42r2 + 3r(9 + 10r) = 1 -18r + 42r2 + (9 * 3r + 10r * 3r) = 1 -18r + 42r2 + (27r + 30r2) = 1 Reorder the terms: -18r + 27r + 42r2 + 30r2 = 1 Combine like terms: -18r + 27r = 9r 9r + 42r2 + 30r2 = 1 Combine like terms: 42r2 + 30r2 = 72r2 9r + 72r2 = 1 Solving 9r + 72r2 = 1 Solving for variable 'r'. Reorder the terms: -1 + 9r + 72r2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 9r + 72r2 = 0 Begin completing the square. Divide all terms by 72 the coefficient of the squared term: Divide each side by '72'. -0.01388888889 + 0.125r + r2 = 0 Move the constant term to the right: Add '0.01388888889' to each side of the equation. -0.01388888889 + 0.125r + 0.01388888889 + r2 = 0 + 0.01388888889 Reorder the terms: -0.01388888889 + 0.01388888889 + 0.125r + r2 = 0 + 0.01388888889 Combine like terms: -0.01388888889 + 0.01388888889 = 0.00000000000 0.00000000000 + 0.125r + r2 = 0 + 0.01388888889 0.125r + r2 = 0 + 0.01388888889 Combine like terms: 0 + 0.01388888889 = 0.01388888889 0.125r + r2 = 0.01388888889 The r term is 0.125r. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125r + 0.00390625 + r2 = 0.01388888889 + 0.00390625 Reorder the terms: 0.00390625 + 0.125r + r2 = 0.01388888889 + 0.00390625 Combine like terms: 0.01388888889 + 0.00390625 = 0.01779513889 0.00390625 + 0.125r + r2 = 0.01779513889 Factor a perfect square on the left side: (r + 0.0625)(r + 0.0625) = 0.01779513889 Calculate the square root of the right side: 0.133398422 Break this problem into two subproblems by setting (r + 0.0625) equal to 0.133398422 and -0.133398422.Subproblem 1
r + 0.0625 = 0.133398422 Simplifying r + 0.0625 = 0.133398422 Reorder the terms: 0.0625 + r = 0.133398422 Solving 0.0625 + r = 0.133398422 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + r = 0.133398422 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + r = 0.133398422 + -0.0625 r = 0.133398422 + -0.0625 Combine like terms: 0.133398422 + -0.0625 = 0.070898422 r = 0.070898422 Simplifying r = 0.070898422Subproblem 2
r + 0.0625 = -0.133398422 Simplifying r + 0.0625 = -0.133398422 Reorder the terms: 0.0625 + r = -0.133398422 Solving 0.0625 + r = -0.133398422 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + r = -0.133398422 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + r = -0.133398422 + -0.0625 r = -0.133398422 + -0.0625 Combine like terms: -0.133398422 + -0.0625 = -0.195898422 r = -0.195898422 Simplifying r = -0.195898422Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.070898422, -0.195898422}
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