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Simplifying 8r3 + 2r2 + -24r = 0 Reorder the terms: -24r + 2r2 + 8r3 = 0 Solving -24r + 2r2 + 8r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2r'. 2r(-12 + r + 4r2) = 0 Ignore the factor 2.Subproblem 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 '(-12 + r + 4r2)' equal to zero and attempt to solve: Simplifying -12 + r + 4r2 = 0 Solving -12 + r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3 + 0.25r + r2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.25r + 3 + r2 = 0 + 3 Reorder the terms: -3 + 3 + 0.25r + r2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.25r + r2 = 0 + 3 0.25r + r2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.25r + r2 = 3 The r term is r. 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. 0.25r + 0.25 + r2 = 3 + 0.25 Reorder the terms: 0.25 + 0.25r + r2 = 3 + 0.25 Combine like terms: 3 + 0.25 = 3.25 0.25 + 0.25r + r2 = 3.25 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (r + 0.5) equal to 1.802775638 and -1.802775638.Subproblem 1
r + 0.5 = 1.802775638 Simplifying r + 0.5 = 1.802775638 Reorder the terms: 0.5 + r = 1.802775638 Solving 0.5 + r = 1.802775638 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 = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.802775638 + -0.5 r = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 r = 1.302775638 Simplifying r = 1.302775638Subproblem 2
r + 0.5 = -1.802775638 Simplifying r + 0.5 = -1.802775638 Reorder the terms: 0.5 + r = -1.802775638 Solving 0.5 + r = -1.802775638 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 = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.802775638 + -0.5 r = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 r = -2.302775638 Simplifying r = -2.302775638Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.302775638, -2.302775638}Solution
r = {0, 1.302775638, -2.302775638}
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