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Simplifying 0 = 5(r2 + 4) + -30r Reorder the terms: 0 = 5(4 + r2) + -30r 0 = (4 * 5 + r2 * 5) + -30r 0 = (20 + 5r2) + -30r Reorder the terms: 0 = 20 + -30r + 5r2 Solving 0 = 20 + -30r + 5r2 Solving for variable 'r'. Combine like terms: 0 + -20 = -20 -20 + 30r + -5r2 = 20 + -30r + 5r2 + -20 + 30r + -5r2 Reorder the terms: -20 + 30r + -5r2 = 20 + -20 + -30r + 30r + 5r2 + -5r2 Combine like terms: 20 + -20 = 0 -20 + 30r + -5r2 = 0 + -30r + 30r + 5r2 + -5r2 -20 + 30r + -5r2 = -30r + 30r + 5r2 + -5r2 Combine like terms: -30r + 30r = 0 -20 + 30r + -5r2 = 0 + 5r2 + -5r2 -20 + 30r + -5r2 = 5r2 + -5r2 Combine like terms: 5r2 + -5r2 = 0 -20 + 30r + -5r2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-4 + 6r + -1r2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-4 + 6r + -1r2)' equal to zero and attempt to solve: Simplifying -4 + 6r + -1r2 = 0 Solving -4 + 6r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 4 + -6r + r2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -6r + -4 + r2 = 0 + -4 Reorder the terms: 4 + -4 + -6r + r2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -6r + r2 = 0 + -4 -6r + r2 = 0 + -4 Combine like terms: 0 + -4 = -4 -6r + r2 = -4 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 = -4 + 9 Reorder the terms: 9 + -6r + r2 = -4 + 9 Combine like terms: -4 + 9 = 5 9 + -6r + r2 = 5 Factor a perfect square on the left side: (r + -3)(r + -3) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (r + -3) equal to 2.236067978 and -2.236067978.Subproblem 1
r + -3 = 2.236067978 Simplifying r + -3 = 2.236067978 Reorder the terms: -3 + r = 2.236067978 Solving -3 + r = 2.236067978 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 = 2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + r = 2.236067978 + 3 r = 2.236067978 + 3 Combine like terms: 2.236067978 + 3 = 5.236067978 r = 5.236067978 Simplifying r = 5.236067978Subproblem 2
r + -3 = -2.236067978 Simplifying r + -3 = -2.236067978 Reorder the terms: -3 + r = -2.236067978 Solving -3 + r = -2.236067978 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 = -2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + r = -2.236067978 + 3 r = -2.236067978 + 3 Combine like terms: -2.236067978 + 3 = 0.763932022 r = 0.763932022 Simplifying r = 0.763932022Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.236067978, 0.763932022}Solution
r = {5.236067978, 0.763932022}
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