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Simplifying 2458.2432 = 2 * 3.14r2 + 2 * 3.14r * 4.2 Multiply 2 * 3.14 2458.2432 = 6.28r2 + 2 * 3.14r * 4.2 Reorder the terms for easier multiplication: 2458.2432 = 6.28r2 + 2 * 3.14 * 4.2r Multiply 2 * 3.14 2458.2432 = 6.28r2 + 6.28 * 4.2r Multiply 6.28 * 4.2 2458.2432 = 6.28r2 + 26.376r Reorder the terms: 2458.2432 = 26.376r + 6.28r2 Solving 2458.2432 = 26.376r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2458.2432 + -26.376r + -6.28r2 = 26.376r + -26.376r + 6.28r2 + -6.28r2 Combine like terms: 26.376r + -26.376r = 0.000 2458.2432 + -26.376r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 2458.2432 + -26.376r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2458.2432 + -26.376r + -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'. -391.44 + 4.2r + r2 = 0 Move the constant term to the right: Add '391.44' to each side of the equation. -391.44 + 4.2r + 391.44 + r2 = 0 + 391.44 Reorder the terms: -391.44 + 391.44 + 4.2r + r2 = 0 + 391.44 Combine like terms: -391.44 + 391.44 = 0.00 0.00 + 4.2r + r2 = 0 + 391.44 4.2r + r2 = 0 + 391.44 Combine like terms: 0 + 391.44 = 391.44 4.2r + r2 = 391.44 The r term is 4.2r. Take half its coefficient (2.1). Square it (4.41) and add it to both sides. Add '4.41' to each side of the equation. 4.2r + 4.41 + r2 = 391.44 + 4.41 Reorder the terms: 4.41 + 4.2r + r2 = 391.44 + 4.41 Combine like terms: 391.44 + 4.41 = 395.85 4.41 + 4.2r + r2 = 395.85 Factor a perfect square on the left side: (r + 2.1)(r + 2.1) = 395.85 Calculate the square root of the right side: 19.895979493 Break this problem into two subproblems by setting (r + 2.1) equal to 19.895979493 and -19.895979493.Subproblem 1
r + 2.1 = 19.895979493 Simplifying r + 2.1 = 19.895979493 Reorder the terms: 2.1 + r = 19.895979493 Solving 2.1 + r = 19.895979493 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + r = 19.895979493 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + r = 19.895979493 + -2.1 r = 19.895979493 + -2.1 Combine like terms: 19.895979493 + -2.1 = 17.795979493 r = 17.795979493 Simplifying r = 17.795979493Subproblem 2
r + 2.1 = -19.895979493 Simplifying r + 2.1 = -19.895979493 Reorder the terms: 2.1 + r = -19.895979493 Solving 2.1 + r = -19.895979493 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + r = -19.895979493 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + r = -19.895979493 + -2.1 r = -19.895979493 + -2.1 Combine like terms: -19.895979493 + -2.1 = -21.995979493 r = -21.995979493 Simplifying r = -21.995979493Solution
The solution to the problem is based on the solutions from the subproblems. r = {17.795979493, -21.995979493}
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