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Simplifying 12r3 + -26r2 + 4r = 2r Reorder the terms: 4r + -26r2 + 12r3 = 2r Solving 4r + -26r2 + 12r3 = 2r Solving for variable 'r'. Reorder the terms: 4r + -2r + -26r2 + 12r3 = 2r + -2r Combine like terms: 4r + -2r = 2r 2r + -26r2 + 12r3 = 2r + -2r Combine like terms: 2r + -2r = 0 2r + -26r2 + 12r3 = 0 Factor out the Greatest Common Factor (GCF), '2r'. 2r(1 + -13r + 6r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r = 0Subproblem 2
Set the factor '(1 + -13r + 6r2)' equal to zero and attempt to solve: Simplifying 1 + -13r + 6r2 = 0 Solving 1 + -13r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + -2.166666667r + r2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + -2.166666667r + -0.1666666667 + r2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + -2.166666667r + r2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + -2.166666667r + r2 = 0 + -0.1666666667 -2.166666667r + r2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 -2.166666667r + r2 = -0.1666666667 The r term is -2.166666667r. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667r + 1.173611113 + r2 = -0.1666666667 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667r + r2 = -0.1666666667 + 1.173611113 Combine like terms: -0.1666666667 + 1.173611113 = 1.0069444463 1.173611113 + -2.166666667r + r2 = 1.0069444463 Factor a perfect square on the left side: (r + -1.083333334)(r + -1.083333334) = 1.0069444463 Calculate the square root of the right side: 1.003466216 Break this problem into two subproblems by setting (r + -1.083333334) equal to 1.003466216 and -1.003466216.Subproblem 1
r + -1.083333334 = 1.003466216 Simplifying r + -1.083333334 = 1.003466216 Reorder the terms: -1.083333334 + r = 1.003466216 Solving -1.083333334 + r = 1.003466216 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + r = 1.003466216 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + r = 1.003466216 + 1.083333334 r = 1.003466216 + 1.083333334 Combine like terms: 1.003466216 + 1.083333334 = 2.08679955 r = 2.08679955 Simplifying r = 2.08679955Subproblem 2
r + -1.083333334 = -1.003466216 Simplifying r + -1.083333334 = -1.003466216 Reorder the terms: -1.083333334 + r = -1.003466216 Solving -1.083333334 + r = -1.003466216 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + r = -1.003466216 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + r = -1.003466216 + 1.083333334 r = -1.003466216 + 1.083333334 Combine like terms: -1.003466216 + 1.083333334 = 0.079867118 r = 0.079867118 Simplifying r = 0.079867118Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.08679955, 0.079867118}Solution
r = {0, 2.08679955, 0.079867118}
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