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Simplifying 25.748 = 2 * 3.14r2 + 2 * 3.14r * 3.1 Multiply 2 * 3.14 25.748 = 6.28r2 + 2 * 3.14r * 3.1 Reorder the terms for easier multiplication: 25.748 = 6.28r2 + 2 * 3.14 * 3.1r Multiply 2 * 3.14 25.748 = 6.28r2 + 6.28 * 3.1r Multiply 6.28 * 3.1 25.748 = 6.28r2 + 19.468r Reorder the terms: 25.748 = 19.468r + 6.28r2 Solving 25.748 = 19.468r + 6.28r2 Solving for variable 'r'. Reorder the terms: 25.748 + -19.468r + -6.28r2 = 19.468r + -19.468r + 6.28r2 + -6.28r2 Combine like terms: 19.468r + -19.468r = 0.000 25.748 + -19.468r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 25.748 + -19.468r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 25.748 + -19.468r + -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'. -4.1 + 3.1r + r2 = 0 Move the constant term to the right: Add '4.1' to each side of the equation. -4.1 + 3.1r + 4.1 + r2 = 0 + 4.1 Reorder the terms: -4.1 + 4.1 + 3.1r + r2 = 0 + 4.1 Combine like terms: -4.1 + 4.1 = 0.0 0.0 + 3.1r + r2 = 0 + 4.1 3.1r + r2 = 0 + 4.1 Combine like terms: 0 + 4.1 = 4.1 3.1r + r2 = 4.1 The r term is 3.1r. Take half its coefficient (1.55). Square it (2.4025) and add it to both sides. Add '2.4025' to each side of the equation. 3.1r + 2.4025 + r2 = 4.1 + 2.4025 Reorder the terms: 2.4025 + 3.1r + r2 = 4.1 + 2.4025 Combine like terms: 4.1 + 2.4025 = 6.5025 2.4025 + 3.1r + r2 = 6.5025 Factor a perfect square on the left side: (r + 1.55)(r + 1.55) = 6.5025 Calculate the square root of the right side: 2.55 Break this problem into two subproblems by setting (r + 1.55) equal to 2.55 and -2.55.Subproblem 1
r + 1.55 = 2.55 Simplifying r + 1.55 = 2.55 Reorder the terms: 1.55 + r = 2.55 Solving 1.55 + r = 2.55 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + r = 2.55 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + r = 2.55 + -1.55 r = 2.55 + -1.55 Combine like terms: 2.55 + -1.55 = 1 r = 1 Simplifying r = 1Subproblem 2
r + 1.55 = -2.55 Simplifying r + 1.55 = -2.55 Reorder the terms: 1.55 + r = -2.55 Solving 1.55 + r = -2.55 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + r = -2.55 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + r = -2.55 + -1.55 r = -2.55 + -1.55 Combine like terms: -2.55 + -1.55 = -4.1 r = -4.1 Simplifying r = -4.1Solution
The solution to the problem is based on the solutions from the subproblems. r = {1, -4.1}
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