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Simplifying 2r(7r2 + -15r + 4) = 0 Reorder the terms: 2r(4 + -15r + 7r2) = 0 (4 * 2r + -15r * 2r + 7r2 * 2r) = 0 (8r + -30r2 + 14r3) = 0 Solving 8r + -30r2 + 14r3 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2r'. 2r(4 + -15r + 7r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'r' equal to zero and attempt to solve: Simplifying r = 0 Solving r = 0 Move all terms containing r to the left, all other terms to the right. Simplifying r = 0Subproblem 2
Set the factor '(4 + -15r + 7r2)' equal to zero and attempt to solve: Simplifying 4 + -15r + 7r2 = 0 Solving 4 + -15r + 7r2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.5714285714 + -2.142857143r + r2 = 0 Move the constant term to the right: Add '-0.5714285714' to each side of the equation. 0.5714285714 + -2.142857143r + -0.5714285714 + r2 = 0 + -0.5714285714 Reorder the terms: 0.5714285714 + -0.5714285714 + -2.142857143r + r2 = 0 + -0.5714285714 Combine like terms: 0.5714285714 + -0.5714285714 = 0.0000000000 0.0000000000 + -2.142857143r + r2 = 0 + -0.5714285714 -2.142857143r + r2 = 0 + -0.5714285714 Combine like terms: 0 + -0.5714285714 = -0.5714285714 -2.142857143r + r2 = -0.5714285714 The r term is -2.142857143r. Take half its coefficient (-1.071428572). Square it (1.147959185) and add it to both sides. Add '1.147959185' to each side of the equation. -2.142857143r + 1.147959185 + r2 = -0.5714285714 + 1.147959185 Reorder the terms: 1.147959185 + -2.142857143r + r2 = -0.5714285714 + 1.147959185 Combine like terms: -0.5714285714 + 1.147959185 = 0.5765306136 1.147959185 + -2.142857143r + r2 = 0.5765306136 Factor a perfect square on the left side: (r + -1.071428572)(r + -1.071428572) = 0.5765306136 Calculate the square root of the right side: 0.75929613 Break this problem into two subproblems by setting (r + -1.071428572) equal to 0.75929613 and -0.75929613.Subproblem 1
r + -1.071428572 = 0.75929613 Simplifying r + -1.071428572 = 0.75929613 Reorder the terms: -1.071428572 + r = 0.75929613 Solving -1.071428572 + r = 0.75929613 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.071428572' to each side of the equation. -1.071428572 + 1.071428572 + r = 0.75929613 + 1.071428572 Combine like terms: -1.071428572 + 1.071428572 = 0.000000000 0.000000000 + r = 0.75929613 + 1.071428572 r = 0.75929613 + 1.071428572 Combine like terms: 0.75929613 + 1.071428572 = 1.830724702 r = 1.830724702 Simplifying r = 1.830724702Subproblem 2
r + -1.071428572 = -0.75929613 Simplifying r + -1.071428572 = -0.75929613 Reorder the terms: -1.071428572 + r = -0.75929613 Solving -1.071428572 + r = -0.75929613 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.071428572' to each side of the equation. -1.071428572 + 1.071428572 + r = -0.75929613 + 1.071428572 Combine like terms: -1.071428572 + 1.071428572 = 0.000000000 0.000000000 + r = -0.75929613 + 1.071428572 r = -0.75929613 + 1.071428572 Combine like terms: -0.75929613 + 1.071428572 = 0.312132442 r = 0.312132442 Simplifying r = 0.312132442Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.830724702, 0.312132442}Solution
r = {0, 1.830724702, 0.312132442}
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