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Simplifying 6r3 + 3r2 + -7r = 0 Reorder the terms: -7r + 3r2 + 6r3 = 0 Solving -7r + 3r2 + 6r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r'. r(-7 + 3r + 6r2) = 0Subproblem 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 '(-7 + 3r + 6r2)' equal to zero and attempt to solve: Simplifying -7 + 3r + 6r2 = 0 Solving -7 + 3r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + 0.5r + r2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 0.5r + 1.166666667 + r2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 0.5r + r2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 0.5r + r2 = 0 + 1.166666667 0.5r + r2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 0.5r + r2 = 1.166666667 The r term is 0.5r. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5r + 0.0625 + r2 = 1.166666667 + 0.0625 Reorder the terms: 0.0625 + 0.5r + r2 = 1.166666667 + 0.0625 Combine like terms: 1.166666667 + 0.0625 = 1.229166667 0.0625 + 0.5r + r2 = 1.229166667 Factor a perfect square on the left side: (r + 0.25)(r + 0.25) = 1.229166667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (r + 0.25) equal to 1.108677891 and -1.108677891.Subproblem 1
r + 0.25 = 1.108677891 Simplifying r + 0.25 = 1.108677891 Reorder the terms: 0.25 + r = 1.108677891 Solving 0.25 + r = 1.108677891 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = 1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 1.108677891 + -0.25 r = 1.108677891 + -0.25 Combine like terms: 1.108677891 + -0.25 = 0.858677891 r = 0.858677891 Simplifying r = 0.858677891Subproblem 2
r + 0.25 = -1.108677891 Simplifying r + 0.25 = -1.108677891 Reorder the terms: 0.25 + r = -1.108677891 Solving 0.25 + r = -1.108677891 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = -1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -1.108677891 + -0.25 r = -1.108677891 + -0.25 Combine like terms: -1.108677891 + -0.25 = -1.358677891 r = -1.358677891 Simplifying r = -1.358677891Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.858677891, -1.358677891}Solution
r = {0, 0.858677891, -1.358677891}
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