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Simplifying 159.3864 = 6.28r + 6.28r2 * 6.7 Reorder the terms for easier multiplication: 159.3864 = 6.28r + 6.28 * 6.7r2 Multiply 6.28 * 6.7 159.3864 = 6.28r + 42.076r2 Solving 159.3864 = 6.28r + 42.076r2 Solving for variable 'r'. Reorder the terms: 159.3864 + -6.28r + -42.076r2 = 6.28r + -6.28r + 42.076r2 + -42.076r2 Combine like terms: 6.28r + -6.28r = 0.00 159.3864 + -6.28r + -42.076r2 = 0.00 + 42.076r2 + -42.076r2 159.3864 + -6.28r + -42.076r2 = 42.076r2 + -42.076r2 Combine like terms: 42.076r2 + -42.076r2 = 0.000 159.3864 + -6.28r + -42.076r2 = 0.000 Begin completing the square. Divide all terms by -42.076 the coefficient of the squared term: Divide each side by '-42.076'. -3.788059701 + 0.1492537313r + r2 = 0 Move the constant term to the right: Add '3.788059701' to each side of the equation. -3.788059701 + 0.1492537313r + 3.788059701 + r2 = 0 + 3.788059701 Reorder the terms: -3.788059701 + 3.788059701 + 0.1492537313r + r2 = 0 + 3.788059701 Combine like terms: -3.788059701 + 3.788059701 = 0.000000000 0.000000000 + 0.1492537313r + r2 = 0 + 3.788059701 0.1492537313r + r2 = 0 + 3.788059701 Combine like terms: 0 + 3.788059701 = 3.788059701 0.1492537313r + r2 = 3.788059701 The r term is 0.1492537313r. Take half its coefficient (0.07462686565). Square it (0.005569169077) and add it to both sides. Add '0.005569169077' to each side of the equation. 0.1492537313r + 0.005569169077 + r2 = 3.788059701 + 0.005569169077 Reorder the terms: 0.005569169077 + 0.1492537313r + r2 = 3.788059701 + 0.005569169077 Combine like terms: 3.788059701 + 0.005569169077 = 3.793628870077 0.005569169077 + 0.1492537313r + r2 = 3.793628870077 Factor a perfect square on the left side: (r + 0.07462686565)(r + 0.07462686565) = 3.793628870077 Calculate the square root of the right side: 1.947724023 Break this problem into two subproblems by setting (r + 0.07462686565) equal to 1.947724023 and -1.947724023.Subproblem 1
r + 0.07462686565 = 1.947724023 Simplifying r + 0.07462686565 = 1.947724023 Reorder the terms: 0.07462686565 + r = 1.947724023 Solving 0.07462686565 + r = 1.947724023 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07462686565' to each side of the equation. 0.07462686565 + -0.07462686565 + r = 1.947724023 + -0.07462686565 Combine like terms: 0.07462686565 + -0.07462686565 = 0.00000000000 0.00000000000 + r = 1.947724023 + -0.07462686565 r = 1.947724023 + -0.07462686565 Combine like terms: 1.947724023 + -0.07462686565 = 1.87309715735 r = 1.87309715735 Simplifying r = 1.87309715735Subproblem 2
r + 0.07462686565 = -1.947724023 Simplifying r + 0.07462686565 = -1.947724023 Reorder the terms: 0.07462686565 + r = -1.947724023 Solving 0.07462686565 + r = -1.947724023 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07462686565' to each side of the equation. 0.07462686565 + -0.07462686565 + r = -1.947724023 + -0.07462686565 Combine like terms: 0.07462686565 + -0.07462686565 = 0.00000000000 0.00000000000 + r = -1.947724023 + -0.07462686565 r = -1.947724023 + -0.07462686565 Combine like terms: -1.947724023 + -0.07462686565 = -2.02235088865 r = -2.02235088865 Simplifying r = -2.02235088865Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.87309715735, -2.02235088865}
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