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Simplifying 21r2 + 3r + -2 = 0 Reorder the terms: -2 + 3r + 21r2 = 0 Solving -2 + 3r + 21r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.09523809524 + 0.1428571429r + r2 = 0 Move the constant term to the right: Add '0.09523809524' to each side of the equation. -0.09523809524 + 0.1428571429r + 0.09523809524 + r2 = 0 + 0.09523809524 Reorder the terms: -0.09523809524 + 0.09523809524 + 0.1428571429r + r2 = 0 + 0.09523809524 Combine like terms: -0.09523809524 + 0.09523809524 = 0.00000000000 0.00000000000 + 0.1428571429r + r2 = 0 + 0.09523809524 0.1428571429r + r2 = 0 + 0.09523809524 Combine like terms: 0 + 0.09523809524 = 0.09523809524 0.1428571429r + r2 = 0.09523809524 The r term is 0.1428571429r. Take half its coefficient (0.07142857145). Square it (0.005102040819) and add it to both sides. Add '0.005102040819' to each side of the equation. 0.1428571429r + 0.005102040819 + r2 = 0.09523809524 + 0.005102040819 Reorder the terms: 0.005102040819 + 0.1428571429r + r2 = 0.09523809524 + 0.005102040819 Combine like terms: 0.09523809524 + 0.005102040819 = 0.100340136059 0.005102040819 + 0.1428571429r + r2 = 0.100340136059 Factor a perfect square on the left side: (r + 0.07142857145)(r + 0.07142857145) = 0.100340136059 Calculate the square root of the right side: 0.316765112 Break this problem into two subproblems by setting (r + 0.07142857145) equal to 0.316765112 and -0.316765112.Subproblem 1
r + 0.07142857145 = 0.316765112 Simplifying r + 0.07142857145 = 0.316765112 Reorder the terms: 0.07142857145 + r = 0.316765112 Solving 0.07142857145 + r = 0.316765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + r = 0.316765112 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + r = 0.316765112 + -0.07142857145 r = 0.316765112 + -0.07142857145 Combine like terms: 0.316765112 + -0.07142857145 = 0.24533654055 r = 0.24533654055 Simplifying r = 0.24533654055Subproblem 2
r + 0.07142857145 = -0.316765112 Simplifying r + 0.07142857145 = -0.316765112 Reorder the terms: 0.07142857145 + r = -0.316765112 Solving 0.07142857145 + r = -0.316765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + r = -0.316765112 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + r = -0.316765112 + -0.07142857145 r = -0.316765112 + -0.07142857145 Combine like terms: -0.316765112 + -0.07142857145 = -0.38819368345 r = -0.38819368345 Simplifying r = -0.38819368345Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.24533654055, -0.38819368345}
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