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Simplifying 14r2 = 5 + -7r Solving 14r2 = 5 + -7r Solving for variable 'r'. Reorder the terms: -5 + 7r + 14r2 = 5 + -7r + -5 + 7r Reorder the terms: -5 + 7r + 14r2 = 5 + -5 + -7r + 7r Combine like terms: 5 + -5 = 0 -5 + 7r + 14r2 = 0 + -7r + 7r -5 + 7r + 14r2 = -7r + 7r Combine like terms: -7r + 7r = 0 -5 + 7r + 14r2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.3571428571 + 0.5r + r2 = 0 Move the constant term to the right: Add '0.3571428571' to each side of the equation. -0.3571428571 + 0.5r + 0.3571428571 + r2 = 0 + 0.3571428571 Reorder the terms: -0.3571428571 + 0.3571428571 + 0.5r + r2 = 0 + 0.3571428571 Combine like terms: -0.3571428571 + 0.3571428571 = 0.0000000000 0.0000000000 + 0.5r + r2 = 0 + 0.3571428571 0.5r + r2 = 0 + 0.3571428571 Combine like terms: 0 + 0.3571428571 = 0.3571428571 0.5r + r2 = 0.3571428571 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 = 0.3571428571 + 0.0625 Reorder the terms: 0.0625 + 0.5r + r2 = 0.3571428571 + 0.0625 Combine like terms: 0.3571428571 + 0.0625 = 0.4196428571 0.0625 + 0.5r + r2 = 0.4196428571 Factor a perfect square on the left side: (r + 0.25)(r + 0.25) = 0.4196428571 Calculate the square root of the right side: 0.64779847 Break this problem into two subproblems by setting (r + 0.25) equal to 0.64779847 and -0.64779847.Subproblem 1
r + 0.25 = 0.64779847 Simplifying r + 0.25 = 0.64779847 Reorder the terms: 0.25 + r = 0.64779847 Solving 0.25 + r = 0.64779847 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 = 0.64779847 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 0.64779847 + -0.25 r = 0.64779847 + -0.25 Combine like terms: 0.64779847 + -0.25 = 0.39779847 r = 0.39779847 Simplifying r = 0.39779847Subproblem 2
r + 0.25 = -0.64779847 Simplifying r + 0.25 = -0.64779847 Reorder the terms: 0.25 + r = -0.64779847 Solving 0.25 + r = -0.64779847 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 = -0.64779847 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -0.64779847 + -0.25 r = -0.64779847 + -0.25 Combine like terms: -0.64779847 + -0.25 = -0.89779847 r = -0.89779847 Simplifying r = -0.89779847Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.39779847, -0.89779847}
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