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Simplifying 6r2 + -19r + -8 = 0 Reorder the terms: -8 + -19r + 6r2 = 0 Solving -8 + -19r + 6r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + -3.166666667r + r2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -3.166666667r + 1.333333333 + r2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -3.166666667r + r2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -3.166666667r + r2 = 0 + 1.333333333 -3.166666667r + r2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -3.166666667r + r2 = 1.333333333 The r term is -3.166666667r. Take half its coefficient (-1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. -3.166666667r + 2.506944447 + r2 = 1.333333333 + 2.506944447 Reorder the terms: 2.506944447 + -3.166666667r + r2 = 1.333333333 + 2.506944447 Combine like terms: 1.333333333 + 2.506944447 = 3.84027778 2.506944447 + -3.166666667r + r2 = 3.84027778 Factor a perfect square on the left side: (r + -1.583333334)(r + -1.583333334) = 3.84027778 Calculate the square root of the right side: 1.95966267 Break this problem into two subproblems by setting (r + -1.583333334) equal to 1.95966267 and -1.95966267.Subproblem 1
r + -1.583333334 = 1.95966267 Simplifying r + -1.583333334 = 1.95966267 Reorder the terms: -1.583333334 + r = 1.95966267 Solving -1.583333334 + r = 1.95966267 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + r = 1.95966267 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + r = 1.95966267 + 1.583333334 r = 1.95966267 + 1.583333334 Combine like terms: 1.95966267 + 1.583333334 = 3.542996004 r = 3.542996004 Simplifying r = 3.542996004Subproblem 2
r + -1.583333334 = -1.95966267 Simplifying r + -1.583333334 = -1.95966267 Reorder the terms: -1.583333334 + r = -1.95966267 Solving -1.583333334 + r = -1.95966267 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + r = -1.95966267 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + r = -1.95966267 + 1.583333334 r = -1.95966267 + 1.583333334 Combine like terms: -1.95966267 + 1.583333334 = -0.376329336 r = -0.376329336 Simplifying r = -0.376329336Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.542996004, -0.376329336}
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