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Simplifying -12r2 + 22r + 48 = 0 Reorder the terms: 48 + 22r + -12r2 = 0 Solving 48 + 22r + -12r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(24 + 11r + -6r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(24 + 11r + -6r2)' equal to zero and attempt to solve: Simplifying 24 + 11r + -6r2 = 0 Solving 24 + 11r + -6r2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -4 + -1.833333333r + r2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -1.833333333r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + -1.833333333r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -1.833333333r + r2 = 0 + 4 -1.833333333r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 -1.833333333r + r2 = 4 The r term is -1.833333333r. Take half its coefficient (-0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. -1.833333333r + 0.8402777775 + r2 = 4 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333r + r2 = 4 + 0.8402777775 Combine like terms: 4 + 0.8402777775 = 4.8402777775 0.8402777775 + -1.833333333r + r2 = 4.8402777775 Factor a perfect square on the left side: (r + -0.9166666665)(r + -0.9166666665) = 4.8402777775 Calculate the square root of the right side: 2.20006313 Break this problem into two subproblems by setting (r + -0.9166666665) equal to 2.20006313 and -2.20006313.Subproblem 1
r + -0.9166666665 = 2.20006313 Simplifying r + -0.9166666665 = 2.20006313 Reorder the terms: -0.9166666665 + r = 2.20006313 Solving -0.9166666665 + r = 2.20006313 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + r = 2.20006313 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + r = 2.20006313 + 0.9166666665 r = 2.20006313 + 0.9166666665 Combine like terms: 2.20006313 + 0.9166666665 = 3.1167297965 r = 3.1167297965 Simplifying r = 3.1167297965Subproblem 2
r + -0.9166666665 = -2.20006313 Simplifying r + -0.9166666665 = -2.20006313 Reorder the terms: -0.9166666665 + r = -2.20006313 Solving -0.9166666665 + r = -2.20006313 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + r = -2.20006313 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + r = -2.20006313 + 0.9166666665 r = -2.20006313 + 0.9166666665 Combine like terms: -2.20006313 + 0.9166666665 = -1.2833964635 r = -1.2833964635 Simplifying r = -1.2833964635Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.1167297965, -1.2833964635}Solution
r = {3.1167297965, -1.2833964635}
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