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Simplifying 6r6 + 3r4 + -7r2 = 0 Reorder the terms: -7r2 + 3r4 + 6r6 = 0 Solving -7r2 + 3r4 + 6r6 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r2'. r2(-7 + 3r2 + 6r4) = 0Subproblem 1
Set the factor 'r2' equal to zero and attempt to solve: Simplifying r2 = 0 Solving r2 = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r2 = 0 Take the square root of each side: r = {0}Subproblem 2
Set the factor '(-7 + 3r2 + 6r4)' equal to zero and attempt to solve: Simplifying -7 + 3r2 + 6r4 = 0 Solving -7 + 3r2 + 6r4 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + 0.5r2 + r4 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 0.5r2 + 1.166666667 + r4 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 0.5r2 + r4 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 0.5r2 + r4 = 0 + 1.166666667 0.5r2 + r4 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 0.5r2 + r4 = 1.166666667 The r term is 0.5r2. 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. 0.5r2 + 0.0625 + r4 = 1.166666667 + 0.0625 Reorder the terms: 0.0625 + 0.5r2 + r4 = 1.166666667 + 0.0625 Combine like terms: 1.166666667 + 0.0625 = 1.229166667 0.0625 + 0.5r2 + r4 = 1.229166667 Factor a perfect square on the left side: (r2 + 0.25)(r2 + 0.25) = 1.229166667 Calculate the square root of the right side: 1.108677891 Break this problem into two subproblems by setting (r2 + 0.25) equal to 1.108677891 and -1.108677891.Subproblem 1
r2 + 0.25 = 1.108677891 Simplifying r2 + 0.25 = 1.108677891 Reorder the terms: 0.25 + r2 = 1.108677891 Solving 0.25 + r2 = 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 + r2 = 1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r2 = 1.108677891 + -0.25 r2 = 1.108677891 + -0.25 Combine like terms: 1.108677891 + -0.25 = 0.858677891 r2 = 0.858677891 Simplifying r2 = 0.858677891 Take the square root of each side: r = {-0.926648742, 0.926648742}Subproblem 2
r2 + 0.25 = -1.108677891 Simplifying r2 + 0.25 = -1.108677891 Reorder the terms: 0.25 + r2 = -1.108677891 Solving 0.25 + r2 = -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 + r2 = -1.108677891 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r2 = -1.108677891 + -0.25 r2 = -1.108677891 + -0.25 Combine like terms: -1.108677891 + -0.25 = -1.358677891 r2 = -1.358677891 Simplifying r2 = -1.358677891 Reorder the terms: 1.358677891 + r2 = -1.358677891 + 1.358677891 Combine like terms: -1.358677891 + 1.358677891 = 0.000000000 1.358677891 + r2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Solution
r = {0}
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