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Simplifying 322.816 = 6.28r2 + 90.432r Reorder the terms: 322.816 = 90.432r + 6.28r2 Solving 322.816 = 90.432r + 6.28r2 Solving for variable 'r'. Reorder the terms: 322.816 + -90.432r + -6.28r2 = 90.432r + -90.432r + 6.28r2 + -6.28r2 Combine like terms: 90.432r + -90.432r = 0.000 322.816 + -90.432r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 322.816 + -90.432r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 322.816 + -90.432r + -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'. -51.40382166 + 14.4r + r2 = 0 Move the constant term to the right: Add '51.40382166' to each side of the equation. -51.40382166 + 14.4r + 51.40382166 + r2 = 0 + 51.40382166 Reorder the terms: -51.40382166 + 51.40382166 + 14.4r + r2 = 0 + 51.40382166 Combine like terms: -51.40382166 + 51.40382166 = 0.00000000 0.00000000 + 14.4r + r2 = 0 + 51.40382166 14.4r + r2 = 0 + 51.40382166 Combine like terms: 0 + 51.40382166 = 51.40382166 14.4r + r2 = 51.40382166 The r term is 14.4r. Take half its coefficient (7.2). Square it (51.84) and add it to both sides. Add '51.84' to each side of the equation. 14.4r + 51.84 + r2 = 51.40382166 + 51.84 Reorder the terms: 51.84 + 14.4r + r2 = 51.40382166 + 51.84 Combine like terms: 51.40382166 + 51.84 = 103.24382166 51.84 + 14.4r + r2 = 103.24382166 Factor a perfect square on the left side: (r + 7.2)(r + 7.2) = 103.24382166 Calculate the square root of the right side: 10.160896696 Break this problem into two subproblems by setting (r + 7.2) equal to 10.160896696 and -10.160896696.Subproblem 1
r + 7.2 = 10.160896696 Simplifying r + 7.2 = 10.160896696 Reorder the terms: 7.2 + r = 10.160896696 Solving 7.2 + r = 10.160896696 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.2' to each side of the equation. 7.2 + -7.2 + r = 10.160896696 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = 10.160896696 + -7.2 r = 10.160896696 + -7.2 Combine like terms: 10.160896696 + -7.2 = 2.960896696 r = 2.960896696 Simplifying r = 2.960896696Subproblem 2
r + 7.2 = -10.160896696 Simplifying r + 7.2 = -10.160896696 Reorder the terms: 7.2 + r = -10.160896696 Solving 7.2 + r = -10.160896696 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.2' to each side of the equation. 7.2 + -7.2 + r = -10.160896696 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = -10.160896696 + -7.2 r = -10.160896696 + -7.2 Combine like terms: -10.160896696 + -7.2 = -17.360896696 r = -17.360896696 Simplifying r = -17.360896696Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.960896696, -17.360896696}
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