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Simplifying 9r2 + 18r + -18 = 6 Reorder the terms: -18 + 18r + 9r2 = 6 Solving -18 + 18r + 9r2 = 6 Solving for variable 'r'. Reorder the terms: -18 + -6 + 18r + 9r2 = 6 + -6 Combine like terms: -18 + -6 = -24 -24 + 18r + 9r2 = 6 + -6 Combine like terms: 6 + -6 = 0 -24 + 18r + 9r2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-8 + 6r + 3r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-8 + 6r + 3r2)' equal to zero and attempt to solve: Simplifying -8 + 6r + 3r2 = 0 Solving -8 + 6r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.666666667 + 2r + r2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + 2r + 2.666666667 + r2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + 2r + r2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + 2r + r2 = 0 + 2.666666667 2r + r2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 2r + r2 = 2.666666667 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 2.666666667 + 1 Reorder the terms: 1 + 2r + r2 = 2.666666667 + 1 Combine like terms: 2.666666667 + 1 = 3.666666667 1 + 2r + r2 = 3.666666667 Factor a perfect square on the left side: (r + 1)(r + 1) = 3.666666667 Calculate the square root of the right side: 1.914854216 Break this problem into two subproblems by setting (r + 1) equal to 1.914854216 and -1.914854216.Subproblem 1
r + 1 = 1.914854216 Simplifying r + 1 = 1.914854216 Reorder the terms: 1 + r = 1.914854216 Solving 1 + r = 1.914854216 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 1.914854216 + -1 Combine like terms: 1 + -1 = 0 0 + r = 1.914854216 + -1 r = 1.914854216 + -1 Combine like terms: 1.914854216 + -1 = 0.914854216 r = 0.914854216 Simplifying r = 0.914854216Subproblem 2
r + 1 = -1.914854216 Simplifying r + 1 = -1.914854216 Reorder the terms: 1 + r = -1.914854216 Solving 1 + r = -1.914854216 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -1.914854216 + -1 Combine like terms: 1 + -1 = 0 0 + r = -1.914854216 + -1 r = -1.914854216 + -1 Combine like terms: -1.914854216 + -1 = -2.914854216 r = -2.914854216 Simplifying r = -2.914854216Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.914854216, -2.914854216}Solution
r = {0.914854216, -2.914854216}
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