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Simplifying 14r2 + 39r + -27 = 0 Reorder the terms: -27 + 39r + 14r2 = 0 Solving -27 + 39r + 14r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.928571429 + 2.785714286r + r2 = 0 Move the constant term to the right: Add '1.928571429' to each side of the equation. -1.928571429 + 2.785714286r + 1.928571429 + r2 = 0 + 1.928571429 Reorder the terms: -1.928571429 + 1.928571429 + 2.785714286r + r2 = 0 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + 2.785714286r + r2 = 0 + 1.928571429 2.785714286r + r2 = 0 + 1.928571429 Combine like terms: 0 + 1.928571429 = 1.928571429 2.785714286r + r2 = 1.928571429 The r term is 2.785714286r. Take half its coefficient (1.392857143). Square it (1.940051021) and add it to both sides. Add '1.940051021' to each side of the equation. 2.785714286r + 1.940051021 + r2 = 1.928571429 + 1.940051021 Reorder the terms: 1.940051021 + 2.785714286r + r2 = 1.928571429 + 1.940051021 Combine like terms: 1.928571429 + 1.940051021 = 3.86862245 1.940051021 + 2.785714286r + r2 = 3.86862245 Factor a perfect square on the left side: (r + 1.392857143)(r + 1.392857143) = 3.86862245 Calculate the square root of the right side: 1.966881402 Break this problem into two subproblems by setting (r + 1.392857143) equal to 1.966881402 and -1.966881402.Subproblem 1
r + 1.392857143 = 1.966881402 Simplifying r + 1.392857143 = 1.966881402 Reorder the terms: 1.392857143 + r = 1.966881402 Solving 1.392857143 + r = 1.966881402 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.392857143' to each side of the equation. 1.392857143 + -1.392857143 + r = 1.966881402 + -1.392857143 Combine like terms: 1.392857143 + -1.392857143 = 0.000000000 0.000000000 + r = 1.966881402 + -1.392857143 r = 1.966881402 + -1.392857143 Combine like terms: 1.966881402 + -1.392857143 = 0.574024259 r = 0.574024259 Simplifying r = 0.574024259Subproblem 2
r + 1.392857143 = -1.966881402 Simplifying r + 1.392857143 = -1.966881402 Reorder the terms: 1.392857143 + r = -1.966881402 Solving 1.392857143 + r = -1.966881402 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.392857143' to each side of the equation. 1.392857143 + -1.392857143 + r = -1.966881402 + -1.392857143 Combine like terms: 1.392857143 + -1.392857143 = 0.000000000 0.000000000 + r = -1.966881402 + -1.392857143 r = -1.966881402 + -1.392857143 Combine like terms: -1.966881402 + -1.392857143 = -3.359738545 r = -3.359738545 Simplifying r = -3.359738545Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.574024259, -3.359738545}
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