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Simplifying r2 + -16r4 + 64r3 = 0 Reorder the terms: r2 + 64r3 + -16r4 = 0 Solving r2 + 64r3 + -16r4 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r2'. r2(1 + 64r + -16r2) = 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 '(1 + 64r + -16r2)' equal to zero and attempt to solve: Simplifying 1 + 64r + -16r2 = 0 Solving 1 + 64r + -16r2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.0625 + -4r + r2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -4r + 0.0625 + r2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -4r + r2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -4r + r2 = 0 + 0.0625 -4r + r2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -4r + r2 = 0.0625 The r term is -4r. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4r + 4 + r2 = 0.0625 + 4 Reorder the terms: 4 + -4r + r2 = 0.0625 + 4 Combine like terms: 0.0625 + 4 = 4.0625 4 + -4r + r2 = 4.0625 Factor a perfect square on the left side: (r + -2)(r + -2) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (r + -2) equal to 2.015564437 and -2.015564437.Subproblem 1
r + -2 = 2.015564437 Simplifying r + -2 = 2.015564437 Reorder the terms: -2 + r = 2.015564437 Solving -2 + r = 2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = 2.015564437 + 2 Combine like terms: -2 + 2 = 0 0 + r = 2.015564437 + 2 r = 2.015564437 + 2 Combine like terms: 2.015564437 + 2 = 4.015564437 r = 4.015564437 Simplifying r = 4.015564437Subproblem 2
r + -2 = -2.015564437 Simplifying r + -2 = -2.015564437 Reorder the terms: -2 + r = -2.015564437 Solving -2 + r = -2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = -2.015564437 + 2 Combine like terms: -2 + 2 = 0 0 + r = -2.015564437 + 2 r = -2.015564437 + 2 Combine like terms: -2.015564437 + 2 = -0.015564437 r = -0.015564437 Simplifying r = -0.015564437Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.015564437, -0.015564437}Solution
r = {0, 4.015564437, -0.015564437}
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