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Simplifying 16r2 + 6r + -5 = 0 Reorder the terms: -5 + 6r + 16r2 = 0 Solving -5 + 6r + 16r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + 0.375r + r2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + 0.375r + 0.3125 + r2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + 0.375r + r2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + 0.375r + r2 = 0 + 0.3125 0.375r + r2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 0.375r + r2 = 0.3125 The r term is 0.375r. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375r + 0.03515625 + r2 = 0.3125 + 0.03515625 Reorder the terms: 0.03515625 + 0.375r + r2 = 0.3125 + 0.03515625 Combine like terms: 0.3125 + 0.03515625 = 0.34765625 0.03515625 + 0.375r + r2 = 0.34765625 Factor a perfect square on the left side: (r + 0.1875)(r + 0.1875) = 0.34765625 Calculate the square root of the right side: 0.589623821 Break this problem into two subproblems by setting (r + 0.1875) equal to 0.589623821 and -0.589623821.Subproblem 1
r + 0.1875 = 0.589623821 Simplifying r + 0.1875 = 0.589623821 Reorder the terms: 0.1875 + r = 0.589623821 Solving 0.1875 + r = 0.589623821 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + r = 0.589623821 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + r = 0.589623821 + -0.1875 r = 0.589623821 + -0.1875 Combine like terms: 0.589623821 + -0.1875 = 0.402123821 r = 0.402123821 Simplifying r = 0.402123821Subproblem 2
r + 0.1875 = -0.589623821 Simplifying r + 0.1875 = -0.589623821 Reorder the terms: 0.1875 + r = -0.589623821 Solving 0.1875 + r = -0.589623821 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + r = -0.589623821 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + r = -0.589623821 + -0.1875 r = -0.589623821 + -0.1875 Combine like terms: -0.589623821 + -0.1875 = -0.777123821 r = -0.777123821 Simplifying r = -0.777123821Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.402123821, -0.777123821}
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