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Simplifying 2r2 + -13r + -40 = 0 Reorder the terms: -40 + -13r + 2r2 = 0 Solving -40 + -13r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -20 + -6.5r + r2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + -6.5r + 20 + r2 = 0 + 20 Reorder the terms: -20 + 20 + -6.5r + r2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -6.5r + r2 = 0 + 20 -6.5r + r2 = 0 + 20 Combine like terms: 0 + 20 = 20 -6.5r + r2 = 20 The r term is -6.5r. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5r + 10.5625 + r2 = 20 + 10.5625 Reorder the terms: 10.5625 + -6.5r + r2 = 20 + 10.5625 Combine like terms: 20 + 10.5625 = 30.5625 10.5625 + -6.5r + r2 = 30.5625 Factor a perfect square on the left side: (r + -3.25)(r + -3.25) = 30.5625 Calculate the square root of the right side: 5.528336097 Break this problem into two subproblems by setting (r + -3.25) equal to 5.528336097 and -5.528336097.Subproblem 1
r + -3.25 = 5.528336097 Simplifying r + -3.25 = 5.528336097 Reorder the terms: -3.25 + r = 5.528336097 Solving -3.25 + r = 5.528336097 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = 5.528336097 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = 5.528336097 + 3.25 r = 5.528336097 + 3.25 Combine like terms: 5.528336097 + 3.25 = 8.778336097 r = 8.778336097 Simplifying r = 8.778336097Subproblem 2
r + -3.25 = -5.528336097 Simplifying r + -3.25 = -5.528336097 Reorder the terms: -3.25 + r = -5.528336097 Solving -3.25 + r = -5.528336097 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + r = -5.528336097 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + r = -5.528336097 + 3.25 r = -5.528336097 + 3.25 Combine like terms: -5.528336097 + 3.25 = -2.278336097 r = -2.278336097 Simplifying r = -2.278336097Solution
The solution to the problem is based on the solutions from the subproblems. r = {8.778336097, -2.278336097}
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