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Simplifying 21r2 + -133r + -20 = 0 Reorder the terms: -20 + -133r + 21r2 = 0 Solving -20 + -133r + 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'. -0.9523809524 + -6.333333333r + r2 = 0 Move the constant term to the right: Add '0.9523809524' to each side of the equation. -0.9523809524 + -6.333333333r + 0.9523809524 + r2 = 0 + 0.9523809524 Reorder the terms: -0.9523809524 + 0.9523809524 + -6.333333333r + r2 = 0 + 0.9523809524 Combine like terms: -0.9523809524 + 0.9523809524 = 0.0000000000 0.0000000000 + -6.333333333r + r2 = 0 + 0.9523809524 -6.333333333r + r2 = 0 + 0.9523809524 Combine like terms: 0 + 0.9523809524 = 0.9523809524 -6.333333333r + r2 = 0.9523809524 The r term is -6.333333333r. Take half its coefficient (-3.166666667). Square it (10.02777778) and add it to both sides. Add '10.02777778' to each side of the equation. -6.333333333r + 10.02777778 + r2 = 0.9523809524 + 10.02777778 Reorder the terms: 10.02777778 + -6.333333333r + r2 = 0.9523809524 + 10.02777778 Combine like terms: 0.9523809524 + 10.02777778 = 10.9801587324 10.02777778 + -6.333333333r + r2 = 10.9801587324 Factor a perfect square on the left side: (r + -3.166666667)(r + -3.166666667) = 10.9801587324 Calculate the square root of the right side: 3.313632257 Break this problem into two subproblems by setting (r + -3.166666667) equal to 3.313632257 and -3.313632257.Subproblem 1
r + -3.166666667 = 3.313632257 Simplifying r + -3.166666667 = 3.313632257 Reorder the terms: -3.166666667 + r = 3.313632257 Solving -3.166666667 + r = 3.313632257 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + r = 3.313632257 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + r = 3.313632257 + 3.166666667 r = 3.313632257 + 3.166666667 Combine like terms: 3.313632257 + 3.166666667 = 6.480298924 r = 6.480298924 Simplifying r = 6.480298924Subproblem 2
r + -3.166666667 = -3.313632257 Simplifying r + -3.166666667 = -3.313632257 Reorder the terms: -3.166666667 + r = -3.313632257 Solving -3.166666667 + r = -3.313632257 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + r = -3.313632257 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + r = -3.313632257 + 3.166666667 r = -3.313632257 + 3.166666667 Combine like terms: -3.313632257 + 3.166666667 = -0.14696559 r = -0.14696559 Simplifying r = -0.14696559Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.480298924, -0.14696559}
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