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Simplifying 965.3616 = 6.28 * 14r2 + 6.28r * 9.9 Multiply 6.28 * 14 965.3616 = 87.92r2 + 6.28r * 9.9 Reorder the terms for easier multiplication: 965.3616 = 87.92r2 + 6.28 * 9.9r Multiply 6.28 * 9.9 965.3616 = 87.92r2 + 62.172r Reorder the terms: 965.3616 = 62.172r + 87.92r2 Solving 965.3616 = 62.172r + 87.92r2 Solving for variable 'r'. Reorder the terms: 965.3616 + -62.172r + -87.92r2 = 62.172r + -62.172r + 87.92r2 + -87.92r2 Combine like terms: 62.172r + -62.172r = 0.000 965.3616 + -62.172r + -87.92r2 = 0.000 + 87.92r2 + -87.92r2 965.3616 + -62.172r + -87.92r2 = 87.92r2 + -87.92r2 Combine like terms: 87.92r2 + -87.92r2 = 0.00 965.3616 + -62.172r + -87.92r2 = 0.00 Begin completing the square. Divide all terms by -87.92 the coefficient of the squared term: Divide each side by '-87.92'. -10.98 + 0.7071428571r + r2 = 0 Move the constant term to the right: Add '10.98' to each side of the equation. -10.98 + 0.7071428571r + 10.98 + r2 = 0 + 10.98 Reorder the terms: -10.98 + 10.98 + 0.7071428571r + r2 = 0 + 10.98 Combine like terms: -10.98 + 10.98 = 0.00 0.00 + 0.7071428571r + r2 = 0 + 10.98 0.7071428571r + r2 = 0 + 10.98 Combine like terms: 0 + 10.98 = 10.98 0.7071428571r + r2 = 10.98 The r term is 0.7071428571r. Take half its coefficient (0.3535714286). Square it (0.1250127551) and add it to both sides. Add '0.1250127551' to each side of the equation. 0.7071428571r + 0.1250127551 + r2 = 10.98 + 0.1250127551 Reorder the terms: 0.1250127551 + 0.7071428571r + r2 = 10.98 + 0.1250127551 Combine like terms: 10.98 + 0.1250127551 = 11.1050127551 0.1250127551 + 0.7071428571r + r2 = 11.1050127551 Factor a perfect square on the left side: (r + 0.3535714286)(r + 0.3535714286) = 11.1050127551 Calculate the square root of the right side: 3.332418454 Break this problem into two subproblems by setting (r + 0.3535714286) equal to 3.332418454 and -3.332418454.Subproblem 1
r + 0.3535714286 = 3.332418454 Simplifying r + 0.3535714286 = 3.332418454 Reorder the terms: 0.3535714286 + r = 3.332418454 Solving 0.3535714286 + r = 3.332418454 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.3535714286' to each side of the equation. 0.3535714286 + -0.3535714286 + r = 3.332418454 + -0.3535714286 Combine like terms: 0.3535714286 + -0.3535714286 = 0.0000000000 0.0000000000 + r = 3.332418454 + -0.3535714286 r = 3.332418454 + -0.3535714286 Combine like terms: 3.332418454 + -0.3535714286 = 2.9788470254 r = 2.9788470254 Simplifying r = 2.9788470254Subproblem 2
r + 0.3535714286 = -3.332418454 Simplifying r + 0.3535714286 = -3.332418454 Reorder the terms: 0.3535714286 + r = -3.332418454 Solving 0.3535714286 + r = -3.332418454 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.3535714286' to each side of the equation. 0.3535714286 + -0.3535714286 + r = -3.332418454 + -0.3535714286 Combine like terms: 0.3535714286 + -0.3535714286 = 0.0000000000 0.0000000000 + r = -3.332418454 + -0.3535714286 r = -3.332418454 + -0.3535714286 Combine like terms: -3.332418454 + -0.3535714286 = -3.6859898826 r = -3.6859898826 Simplifying r = -3.6859898826Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.9788470254, -3.6859898826}
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