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Simplifying 7r2 + 11r + -9 = 0 Reorder the terms: -9 + 11r + 7r2 = 0 Solving -9 + 11r + 7r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.285714286 + 1.571428571r + r2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 1.571428571r + 1.285714286 + r2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 1.571428571r + r2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 1.571428571r + r2 = 0 + 1.285714286 1.571428571r + r2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 1.571428571r + r2 = 1.285714286 The r term is 1.571428571r. Take half its coefficient (0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. 1.571428571r + 0.6173469384 + r2 = 1.285714286 + 0.6173469384 Reorder the terms: 0.6173469384 + 1.571428571r + r2 = 1.285714286 + 0.6173469384 Combine like terms: 1.285714286 + 0.6173469384 = 1.9030612244 0.6173469384 + 1.571428571r + r2 = 1.9030612244 Factor a perfect square on the left side: (r + 0.7857142855)(r + 0.7857142855) = 1.9030612244 Calculate the square root of the right side: 1.379514851 Break this problem into two subproblems by setting (r + 0.7857142855) equal to 1.379514851 and -1.379514851.Subproblem 1
r + 0.7857142855 = 1.379514851 Simplifying r + 0.7857142855 = 1.379514851 Reorder the terms: 0.7857142855 + r = 1.379514851 Solving 0.7857142855 + r = 1.379514851 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + r = 1.379514851 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + r = 1.379514851 + -0.7857142855 r = 1.379514851 + -0.7857142855 Combine like terms: 1.379514851 + -0.7857142855 = 0.5938005655 r = 0.5938005655 Simplifying r = 0.5938005655Subproblem 2
r + 0.7857142855 = -1.379514851 Simplifying r + 0.7857142855 = -1.379514851 Reorder the terms: 0.7857142855 + r = -1.379514851 Solving 0.7857142855 + r = -1.379514851 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + r = -1.379514851 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + r = -1.379514851 + -0.7857142855 r = -1.379514851 + -0.7857142855 Combine like terms: -1.379514851 + -0.7857142855 = -2.1652291365 r = -2.1652291365 Simplifying r = -2.1652291365Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.5938005655, -2.1652291365}
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