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Simplifying 18r2 + -22r + -42 = 0 Reorder the terms: -42 + -22r + 18r2 = 0 Solving -42 + -22r + 18r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-21 + -11r + 9r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-21 + -11r + 9r2)' equal to zero and attempt to solve: Simplifying -21 + -11r + 9r2 = 0 Solving -21 + -11r + 9r2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.333333333 + -1.222222222r + r2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -1.222222222r + 2.333333333 + r2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -1.222222222r + r2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -1.222222222r + r2 = 0 + 2.333333333 -1.222222222r + r2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -1.222222222r + r2 = 2.333333333 The r term is -1.222222222r. Take half its coefficient (-0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. -1.222222222r + 0.3734567900 + r2 = 2.333333333 + 0.3734567900 Reorder the terms: 0.3734567900 + -1.222222222r + r2 = 2.333333333 + 0.3734567900 Combine like terms: 2.333333333 + 0.3734567900 = 2.706790123 0.3734567900 + -1.222222222r + r2 = 2.706790123 Factor a perfect square on the left side: (r + -0.611111111)(r + -0.611111111) = 2.706790123 Calculate the square root of the right side: 1.645232544 Break this problem into two subproblems by setting (r + -0.611111111) equal to 1.645232544 and -1.645232544.Subproblem 1
r + -0.611111111 = 1.645232544 Simplifying r + -0.611111111 = 1.645232544 Reorder the terms: -0.611111111 + r = 1.645232544 Solving -0.611111111 + r = 1.645232544 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = 1.645232544 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = 1.645232544 + 0.611111111 r = 1.645232544 + 0.611111111 Combine like terms: 1.645232544 + 0.611111111 = 2.256343655 r = 2.256343655 Simplifying r = 2.256343655Subproblem 2
r + -0.611111111 = -1.645232544 Simplifying r + -0.611111111 = -1.645232544 Reorder the terms: -0.611111111 + r = -1.645232544 Solving -0.611111111 + r = -1.645232544 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = -1.645232544 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = -1.645232544 + 0.611111111 r = -1.645232544 + 0.611111111 Combine like terms: -1.645232544 + 0.611111111 = -1.034121433 r = -1.034121433 Simplifying r = -1.034121433Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.256343655, -1.034121433}Solution
r = {2.256343655, -1.034121433}
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