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Simplifying r3 + -2r2 + -4r = 0 Reorder the terms: -4r + -2r2 + r3 = 0 Solving -4r + -2r2 + r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), 'r'. r(-4 + -2r + r2) = 0Subproblem 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 '(-4 + -2r + r2)' equal to zero and attempt to solve: Simplifying -4 + -2r + r2 = 0 Solving -4 + -2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + -2r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + -2r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2r + r2 = 0 + 4 -2r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2r + r2 = 4 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = 4 + 1 Reorder the terms: 1 + -2r + r2 = 4 + 1 Combine like terms: 4 + 1 = 5 1 + -2r + r2 = 5 Factor a perfect square on the left side: (r + -1)(r + -1) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (r + -1) equal to 2.236067978 and -2.236067978.Subproblem 1
r + -1 = 2.236067978 Simplifying r + -1 = 2.236067978 Reorder the terms: -1 + r = 2.236067978 Solving -1 + r = 2.236067978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + r = 2.236067978 + 1 r = 2.236067978 + 1 Combine like terms: 2.236067978 + 1 = 3.236067978 r = 3.236067978 Simplifying r = 3.236067978Subproblem 2
r + -1 = -2.236067978 Simplifying r + -1 = -2.236067978 Reorder the terms: -1 + r = -2.236067978 Solving -1 + r = -2.236067978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + r = -2.236067978 + 1 r = -2.236067978 + 1 Combine like terms: -2.236067978 + 1 = -1.236067978 r = -1.236067978 Simplifying r = -1.236067978Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.236067978, -1.236067978}Solution
r = {0, 3.236067978, -1.236067978}
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