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Simplifying 15r3 + 6r2 + -60r = 0 Reorder the terms: -60r + 6r2 + 15r3 = 0 Solving -60r + 6r2 + 15r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3r'. 3r(-20 + 2r + 5r2) = 0 Ignore the factor 3.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 '(-20 + 2r + 5r2)' equal to zero and attempt to solve: Simplifying -20 + 2r + 5r2 = 0 Solving -20 + 2r + 5r2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4 + 0.4r + r2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.4r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + 0.4r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.4r + r2 = 0 + 4 0.4r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.4r + r2 = 4 The r term is 0.4r. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4r + 0.04 + r2 = 4 + 0.04 Reorder the terms: 0.04 + 0.4r + r2 = 4 + 0.04 Combine like terms: 4 + 0.04 = 4.04 0.04 + 0.4r + r2 = 4.04 Factor a perfect square on the left side: (r + 0.2)(r + 0.2) = 4.04 Calculate the square root of the right side: 2.009975124 Break this problem into two subproblems by setting (r + 0.2) equal to 2.009975124 and -2.009975124.Subproblem 1
r + 0.2 = 2.009975124 Simplifying r + 0.2 = 2.009975124 Reorder the terms: 0.2 + r = 2.009975124 Solving 0.2 + r = 2.009975124 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + r = 2.009975124 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + r = 2.009975124 + -0.2 r = 2.009975124 + -0.2 Combine like terms: 2.009975124 + -0.2 = 1.809975124 r = 1.809975124 Simplifying r = 1.809975124Subproblem 2
r + 0.2 = -2.009975124 Simplifying r + 0.2 = -2.009975124 Reorder the terms: 0.2 + r = -2.009975124 Solving 0.2 + r = -2.009975124 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + r = -2.009975124 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + r = -2.009975124 + -0.2 r = -2.009975124 + -0.2 Combine like terms: -2.009975124 + -0.2 = -2.209975124 r = -2.209975124 Simplifying r = -2.209975124Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.809975124, -2.209975124}Solution
r = {0, 1.809975124, -2.209975124}
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