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Simplifying 12 = 6.28r(r + 5) Reorder the terms: 12 = 6.28r(5 + r) 12 = (5 * 6.28r + r * 6.28r) 12 = (31.4r + 6.28r2) Solving 12 = 31.4r + 6.28r2 Solving for variable 'r'. Reorder the terms: 12 + -31.4r + -6.28r2 = 31.4r + -31.4r + 6.28r2 + -6.28r2 Combine like terms: 31.4r + -31.4r = 0.0 12 + -31.4r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 12 + -31.4r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 12 + -31.4r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -1.910828025 + 5r + r2 = 0 Move the constant term to the right: Add '1.910828025' to each side of the equation. -1.910828025 + 5r + 1.910828025 + r2 = 0 + 1.910828025 Reorder the terms: -1.910828025 + 1.910828025 + 5r + r2 = 0 + 1.910828025 Combine like terms: -1.910828025 + 1.910828025 = 0.000000000 0.000000000 + 5r + r2 = 0 + 1.910828025 5r + r2 = 0 + 1.910828025 Combine like terms: 0 + 1.910828025 = 1.910828025 5r + r2 = 1.910828025 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = 1.910828025 + 6.25 Reorder the terms: 6.25 + 5r + r2 = 1.910828025 + 6.25 Combine like terms: 1.910828025 + 6.25 = 8.160828025 6.25 + 5r + r2 = 8.160828025 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 8.160828025 Calculate the square root of the right side: 2.856716301 Break this problem into two subproblems by setting (r + 2.5) equal to 2.856716301 and -2.856716301.Subproblem 1
r + 2.5 = 2.856716301 Simplifying r + 2.5 = 2.856716301 Reorder the terms: 2.5 + r = 2.856716301 Solving 2.5 + r = 2.856716301 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 2.856716301 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 2.856716301 + -2.5 r = 2.856716301 + -2.5 Combine like terms: 2.856716301 + -2.5 = 0.356716301 r = 0.356716301 Simplifying r = 0.356716301Subproblem 2
r + 2.5 = -2.856716301 Simplifying r + 2.5 = -2.856716301 Reorder the terms: 2.5 + r = -2.856716301 Solving 2.5 + r = -2.856716301 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -2.856716301 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -2.856716301 + -2.5 r = -2.856716301 + -2.5 Combine like terms: -2.856716301 + -2.5 = -5.356716301 r = -5.356716301 Simplifying r = -5.356716301Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.356716301, -5.356716301}
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