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Simplifying r4 = r3 + 21r2 Reorder the terms: r4 = 21r2 + r3 Solving r4 = 21r2 + r3 Solving for variable 'r'. Reorder the terms: -21r2 + -1r3 + r4 = 21r2 + r3 + -21r2 + -1r3 Reorder the terms: -21r2 + -1r3 + r4 = 21r2 + -21r2 + r3 + -1r3 Combine like terms: 21r2 + -21r2 = 0 -21r2 + -1r3 + r4 = 0 + r3 + -1r3 -21r2 + -1r3 + r4 = r3 + -1r3 Combine like terms: r3 + -1r3 = 0 -21r2 + -1r3 + r4 = 0 Factor out the Greatest Common Factor (GCF), 'r2'. r2(-21 + -1r + 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 '(-21 + -1r + r2)' equal to zero and attempt to solve: Simplifying -21 + -1r + r2 = 0 Solving -21 + -1r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + -1r + 21 + r2 = 0 + 21 Reorder the terms: -21 + 21 + -1r + r2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + -1r + r2 = 0 + 21 -1r + r2 = 0 + 21 Combine like terms: 0 + 21 = 21 -1r + r2 = 21 The r term is -1r. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1r + 0.25 + r2 = 21 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 21 + 0.25 Combine like terms: 21 + 0.25 = 21.25 0.25 + -1r + r2 = 21.25 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 21.25 Calculate the square root of the right side: 4.609772229 Break this problem into two subproblems by setting (r + -0.5) equal to 4.609772229 and -4.609772229.Subproblem 1
r + -0.5 = 4.609772229 Simplifying r + -0.5 = 4.609772229 Reorder the terms: -0.5 + r = 4.609772229 Solving -0.5 + r = 4.609772229 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = 4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 4.609772229 + 0.5 r = 4.609772229 + 0.5 Combine like terms: 4.609772229 + 0.5 = 5.109772229 r = 5.109772229 Simplifying r = 5.109772229Subproblem 2
r + -0.5 = -4.609772229 Simplifying r + -0.5 = -4.609772229 Reorder the terms: -0.5 + r = -4.609772229 Solving -0.5 + r = -4.609772229 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + r = -4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -4.609772229 + 0.5 r = -4.609772229 + 0.5 Combine like terms: -4.609772229 + 0.5 = -4.109772229 r = -4.109772229 Simplifying r = -4.109772229Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.109772229, -4.109772229}Solution
r = {0, 5.109772229, -4.109772229}
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