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Simplifying 3r2 + -8r + -9 = 0 Reorder the terms: -9 + -8r + 3r2 = 0 Solving -9 + -8r + 3r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3 + -2.666666667r + r2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -2.666666667r + 3 + r2 = 0 + 3 Reorder the terms: -3 + 3 + -2.666666667r + r2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -2.666666667r + r2 = 0 + 3 -2.666666667r + r2 = 0 + 3 Combine like terms: 0 + 3 = 3 -2.666666667r + r2 = 3 The r term is -2.666666667r. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667r + 1.777777780 + r2 = 3 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667r + r2 = 3 + 1.777777780 Combine like terms: 3 + 1.777777780 = 4.77777778 1.777777780 + -2.666666667r + r2 = 4.77777778 Factor a perfect square on the left side: (r + -1.333333334)(r + -1.333333334) = 4.77777778 Calculate the square root of the right side: 2.185812842 Break this problem into two subproblems by setting (r + -1.333333334) equal to 2.185812842 and -2.185812842.Subproblem 1
r + -1.333333334 = 2.185812842 Simplifying r + -1.333333334 = 2.185812842 Reorder the terms: -1.333333334 + r = 2.185812842 Solving -1.333333334 + r = 2.185812842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = 2.185812842 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = 2.185812842 + 1.333333334 r = 2.185812842 + 1.333333334 Combine like terms: 2.185812842 + 1.333333334 = 3.519146176 r = 3.519146176 Simplifying r = 3.519146176Subproblem 2
r + -1.333333334 = -2.185812842 Simplifying r + -1.333333334 = -2.185812842 Reorder the terms: -1.333333334 + r = -2.185812842 Solving -1.333333334 + r = -2.185812842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = -2.185812842 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = -2.185812842 + 1.333333334 r = -2.185812842 + 1.333333334 Combine like terms: -2.185812842 + 1.333333334 = -0.852479508 r = -0.852479508 Simplifying r = -0.852479508Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.519146176, -0.852479508}
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