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Simplifying 3r + 9r(12 + -2r) = 2 3r + (12 * 9r + -2r * 9r) = 2 3r + (108r + -18r2) = 2 Combine like terms: 3r + 108r = 111r 111r + -18r2 = 2 Solving 111r + -18r2 = 2 Solving for variable 'r'. Reorder the terms: -2 + 111r + -18r2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + 111r + -18r2 = 0 Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. 0.1111111111 + -6.166666667r + r2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + -6.166666667r + -0.1111111111 + r2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + -6.166666667r + r2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + -6.166666667r + r2 = 0 + -0.1111111111 -6.166666667r + r2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 -6.166666667r + r2 = -0.1111111111 The r term is -6.166666667r. Take half its coefficient (-3.083333334). Square it (9.506944449) and add it to both sides. Add '9.506944449' to each side of the equation. -6.166666667r + 9.506944449 + r2 = -0.1111111111 + 9.506944449 Reorder the terms: 9.506944449 + -6.166666667r + r2 = -0.1111111111 + 9.506944449 Combine like terms: -0.1111111111 + 9.506944449 = 9.3958333379 9.506944449 + -6.166666667r + r2 = 9.3958333379 Factor a perfect square on the left side: (r + -3.083333334)(r + -3.083333334) = 9.3958333379 Calculate the square root of the right side: 3.06526236 Break this problem into two subproblems by setting (r + -3.083333334) equal to 3.06526236 and -3.06526236.Subproblem 1
r + -3.083333334 = 3.06526236 Simplifying r + -3.083333334 = 3.06526236 Reorder the terms: -3.083333334 + r = 3.06526236 Solving -3.083333334 + r = 3.06526236 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.083333334' to each side of the equation. -3.083333334 + 3.083333334 + r = 3.06526236 + 3.083333334 Combine like terms: -3.083333334 + 3.083333334 = 0.000000000 0.000000000 + r = 3.06526236 + 3.083333334 r = 3.06526236 + 3.083333334 Combine like terms: 3.06526236 + 3.083333334 = 6.148595694 r = 6.148595694 Simplifying r = 6.148595694Subproblem 2
r + -3.083333334 = -3.06526236 Simplifying r + -3.083333334 = -3.06526236 Reorder the terms: -3.083333334 + r = -3.06526236 Solving -3.083333334 + r = -3.06526236 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.083333334' to each side of the equation. -3.083333334 + 3.083333334 + r = -3.06526236 + 3.083333334 Combine like terms: -3.083333334 + 3.083333334 = 0.000000000 0.000000000 + r = -3.06526236 + 3.083333334 r = -3.06526236 + 3.083333334 Combine like terms: -3.06526236 + 3.083333334 = 0.018070974 r = 0.018070974 Simplifying r = 0.018070974Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.148595694, 0.018070974}
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