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Simplifying r4 = 6r3 + 24r2 Reorder the terms: r4 = 24r2 + 6r3 Solving r4 = 24r2 + 6r3 Solving for variable 'r'. Reorder the terms: -24r2 + -6r3 + r4 = 24r2 + 6r3 + -24r2 + -6r3 Reorder the terms: -24r2 + -6r3 + r4 = 24r2 + -24r2 + 6r3 + -6r3 Combine like terms: 24r2 + -24r2 = 0 -24r2 + -6r3 + r4 = 0 + 6r3 + -6r3 -24r2 + -6r3 + r4 = 6r3 + -6r3 Combine like terms: 6r3 + -6r3 = 0 -24r2 + -6r3 + r4 = 0 Factor out the Greatest Common Factor (GCF), 'r2'. r2(-24 + -6r + r2) = 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 '(-24 + -6r + r2)' equal to zero and attempt to solve: Simplifying -24 + -6r + r2 = 0 Solving -24 + -6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '24' to each side of the equation. -24 + -6r + 24 + r2 = 0 + 24 Reorder the terms: -24 + 24 + -6r + r2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -6r + r2 = 0 + 24 -6r + r2 = 0 + 24 Combine like terms: 0 + 24 = 24 -6r + r2 = 24 The r term is -6r. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6r + 9 + r2 = 24 + 9 Reorder the terms: 9 + -6r + r2 = 24 + 9 Combine like terms: 24 + 9 = 33 9 + -6r + r2 = 33 Factor a perfect square on the left side: (r + -3)(r + -3) = 33 Calculate the square root of the right side: 5.744562647 Break this problem into two subproblems by setting (r + -3) equal to 5.744562647 and -5.744562647.Subproblem 1
r + -3 = 5.744562647 Simplifying r + -3 = 5.744562647 Reorder the terms: -3 + r = 5.744562647 Solving -3 + r = 5.744562647 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + r = 5.744562647 + 3 Combine like terms: -3 + 3 = 0 0 + r = 5.744562647 + 3 r = 5.744562647 + 3 Combine like terms: 5.744562647 + 3 = 8.744562647 r = 8.744562647 Simplifying r = 8.744562647Subproblem 2
r + -3 = -5.744562647 Simplifying r + -3 = -5.744562647 Reorder the terms: -3 + r = -5.744562647 Solving -3 + r = -5.744562647 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + r = -5.744562647 + 3 Combine like terms: -3 + 3 = 0 0 + r = -5.744562647 + 3 r = -5.744562647 + 3 Combine like terms: -5.744562647 + 3 = -2.744562647 r = -2.744562647 Simplifying r = -2.744562647Solution
The solution to the problem is based on the solutions from the subproblems. r = {8.744562647, -2.744562647}Solution
r = {0, 8.744562647, -2.744562647}
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