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Simplifying 196r2 + -140r + 14 = 0 Reorder the terms: 14 + -140r + 196r2 = 0 Solving 14 + -140r + 196r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '14'. 14(1 + -10r + 14r2) = 0 Ignore the factor 14.Subproblem 1
Set the factor '(1 + -10r + 14r2)' equal to zero and attempt to solve: Simplifying 1 + -10r + 14r2 = 0 Solving 1 + -10r + 14r2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 0.07142857143 + -0.7142857143r + r2 = 0 Move the constant term to the right: Add '-0.07142857143' to each side of the equation. 0.07142857143 + -0.7142857143r + -0.07142857143 + r2 = 0 + -0.07142857143 Reorder the terms: 0.07142857143 + -0.07142857143 + -0.7142857143r + r2 = 0 + -0.07142857143 Combine like terms: 0.07142857143 + -0.07142857143 = 0.00000000000 0.00000000000 + -0.7142857143r + r2 = 0 + -0.07142857143 -0.7142857143r + r2 = 0 + -0.07142857143 Combine like terms: 0 + -0.07142857143 = -0.07142857143 -0.7142857143r + r2 = -0.07142857143 The r term is -0.7142857143r. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143r + 0.1275510204 + r2 = -0.07142857143 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143r + r2 = -0.07142857143 + 0.1275510204 Combine like terms: -0.07142857143 + 0.1275510204 = 0.05612244897 0.1275510204 + -0.7142857143r + r2 = 0.05612244897 Factor a perfect square on the left side: (r + -0.3571428572)(r + -0.3571428572) = 0.05612244897 Calculate the square root of the right side: 0.236901771 Break this problem into two subproblems by setting (r + -0.3571428572) equal to 0.236901771 and -0.236901771.Subproblem 1
r + -0.3571428572 = 0.236901771 Simplifying r + -0.3571428572 = 0.236901771 Reorder the terms: -0.3571428572 + r = 0.236901771 Solving -0.3571428572 + r = 0.236901771 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + r = 0.236901771 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + r = 0.236901771 + 0.3571428572 r = 0.236901771 + 0.3571428572 Combine like terms: 0.236901771 + 0.3571428572 = 0.5940446282 r = 0.5940446282 Simplifying r = 0.5940446282Subproblem 2
r + -0.3571428572 = -0.236901771 Simplifying r + -0.3571428572 = -0.236901771 Reorder the terms: -0.3571428572 + r = -0.236901771 Solving -0.3571428572 + r = -0.236901771 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + r = -0.236901771 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + r = -0.236901771 + 0.3571428572 r = -0.236901771 + 0.3571428572 Combine like terms: -0.236901771 + 0.3571428572 = 0.1202410862 r = 0.1202410862 Simplifying r = 0.1202410862Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.5940446282, 0.1202410862}Solution
r = {0.5940446282, 0.1202410862}
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