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Simplifying 16r2 + 12r + -17 = 8r + 8r2 Reorder the terms: -17 + 12r + 16r2 = 8r + 8r2 Solving -17 + 12r + 16r2 = 8r + 8r2 Solving for variable 'r'. Reorder the terms: -17 + 12r + -8r + 16r2 + -8r2 = 8r + 8r2 + -8r + -8r2 Combine like terms: 12r + -8r = 4r -17 + 4r + 16r2 + -8r2 = 8r + 8r2 + -8r + -8r2 Combine like terms: 16r2 + -8r2 = 8r2 -17 + 4r + 8r2 = 8r + 8r2 + -8r + -8r2 Reorder the terms: -17 + 4r + 8r2 = 8r + -8r + 8r2 + -8r2 Combine like terms: 8r + -8r = 0 -17 + 4r + 8r2 = 0 + 8r2 + -8r2 -17 + 4r + 8r2 = 8r2 + -8r2 Combine like terms: 8r2 + -8r2 = 0 -17 + 4r + 8r2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.125 + 0.5r + r2 = 0 Move the constant term to the right: Add '2.125' to each side of the equation. -2.125 + 0.5r + 2.125 + r2 = 0 + 2.125 Reorder the terms: -2.125 + 2.125 + 0.5r + r2 = 0 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + 0.5r + r2 = 0 + 2.125 0.5r + r2 = 0 + 2.125 Combine like terms: 0 + 2.125 = 2.125 0.5r + r2 = 2.125 The r term is 0.5r. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5r + 0.0625 + r2 = 2.125 + 0.0625 Reorder the terms: 0.0625 + 0.5r + r2 = 2.125 + 0.0625 Combine like terms: 2.125 + 0.0625 = 2.1875 0.0625 + 0.5r + r2 = 2.1875 Factor a perfect square on the left side: (r + 0.25)(r + 0.25) = 2.1875 Calculate the square root of the right side: 1.479019946 Break this problem into two subproblems by setting (r + 0.25) equal to 1.479019946 and -1.479019946.Subproblem 1
r + 0.25 = 1.479019946 Simplifying r + 0.25 = 1.479019946 Reorder the terms: 0.25 + r = 1.479019946 Solving 0.25 + r = 1.479019946 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = 1.479019946 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 1.479019946 + -0.25 r = 1.479019946 + -0.25 Combine like terms: 1.479019946 + -0.25 = 1.229019946 r = 1.229019946 Simplifying r = 1.229019946Subproblem 2
r + 0.25 = -1.479019946 Simplifying r + 0.25 = -1.479019946 Reorder the terms: 0.25 + r = -1.479019946 Solving 0.25 + r = -1.479019946 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = -1.479019946 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -1.479019946 + -0.25 r = -1.479019946 + -0.25 Combine like terms: -1.479019946 + -0.25 = -1.729019946 r = -1.729019946 Simplifying r = -1.729019946Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.229019946, -1.729019946}
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