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Simplifying 2r2 + 3r + -176 = 0 Reorder the terms: -176 + 3r + 2r2 = 0 Solving -176 + 3r + 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'. -88 + 1.5r + r2 = 0 Move the constant term to the right: Add '88' to each side of the equation. -88 + 1.5r + 88 + r2 = 0 + 88 Reorder the terms: -88 + 88 + 1.5r + r2 = 0 + 88 Combine like terms: -88 + 88 = 0 0 + 1.5r + r2 = 0 + 88 1.5r + r2 = 0 + 88 Combine like terms: 0 + 88 = 88 1.5r + r2 = 88 The r term is 1.5r. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5r + 0.5625 + r2 = 88 + 0.5625 Reorder the terms: 0.5625 + 1.5r + r2 = 88 + 0.5625 Combine like terms: 88 + 0.5625 = 88.5625 0.5625 + 1.5r + r2 = 88.5625 Factor a perfect square on the left side: (r + 0.75)(r + 0.75) = 88.5625 Calculate the square root of the right side: 9.410765112 Break this problem into two subproblems by setting (r + 0.75) equal to 9.410765112 and -9.410765112.Subproblem 1
r + 0.75 = 9.410765112 Simplifying r + 0.75 = 9.410765112 Reorder the terms: 0.75 + r = 9.410765112 Solving 0.75 + r = 9.410765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = 9.410765112 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 9.410765112 + -0.75 r = 9.410765112 + -0.75 Combine like terms: 9.410765112 + -0.75 = 8.660765112 r = 8.660765112 Simplifying r = 8.660765112Subproblem 2
r + 0.75 = -9.410765112 Simplifying r + 0.75 = -9.410765112 Reorder the terms: 0.75 + r = -9.410765112 Solving 0.75 + r = -9.410765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = -9.410765112 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -9.410765112 + -0.75 r = -9.410765112 + -0.75 Combine like terms: -9.410765112 + -0.75 = -10.160765112 r = -10.160765112 Simplifying r = -10.160765112Solution
The solution to the problem is based on the solutions from the subproblems. r = {8.660765112, -10.160765112}
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