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Simplifying 6r2 + 19r = 10 Reorder the terms: 19r + 6r2 = 10 Solving 19r + 6r2 = 10 Solving for variable 'r'. Reorder the terms: -10 + 19r + 6r2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 19r + 6r2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.666666667 + 3.166666667r + r2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 3.166666667r + 1.666666667 + r2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 3.166666667r + r2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 3.166666667r + r2 = 0 + 1.666666667 3.166666667r + r2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 3.166666667r + r2 = 1.666666667 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 = 1.666666667 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667r + r2 = 1.666666667 + 2.506944447 Combine like terms: 1.666666667 + 2.506944447 = 4.173611114 2.506944447 + 3.166666667r + r2 = 4.173611114 Factor a perfect square on the left side: (r + 1.583333334)(r + 1.583333334) = 4.173611114 Calculate the square root of the right side: 2.042941779 Break this problem into two subproblems by setting (r + 1.583333334) equal to 2.042941779 and -2.042941779.Subproblem 1
r + 1.583333334 = 2.042941779 Simplifying r + 1.583333334 = 2.042941779 Reorder the terms: 1.583333334 + r = 2.042941779 Solving 1.583333334 + r = 2.042941779 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 = 2.042941779 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + r = 2.042941779 + -1.583333334 r = 2.042941779 + -1.583333334 Combine like terms: 2.042941779 + -1.583333334 = 0.459608445 r = 0.459608445 Simplifying r = 0.459608445Subproblem 2
r + 1.583333334 = -2.042941779 Simplifying r + 1.583333334 = -2.042941779 Reorder the terms: 1.583333334 + r = -2.042941779 Solving 1.583333334 + r = -2.042941779 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 = -2.042941779 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + r = -2.042941779 + -1.583333334 r = -2.042941779 + -1.583333334 Combine like terms: -2.042941779 + -1.583333334 = -3.626275113 r = -3.626275113 Simplifying r = -3.626275113Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.459608445, -3.626275113}
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