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Simplifying 29 + 9r = 11r(-7r + -11) Reorder the terms: 29 + 9r = 11r(-11 + -7r) 29 + 9r = (-11 * 11r + -7r * 11r) 29 + 9r = (-121r + -77r2) Solving 29 + 9r = -121r + -77r2 Solving for variable 'r'. Combine like terms: 9r + 121r = 130r 29 + 130r + 77r2 = -121r + -77r2 + 121r + 77r2 Reorder the terms: 29 + 130r + 77r2 = -121r + 121r + -77r2 + 77r2 Combine like terms: -121r + 121r = 0 29 + 130r + 77r2 = 0 + -77r2 + 77r2 29 + 130r + 77r2 = -77r2 + 77r2 Combine like terms: -77r2 + 77r2 = 0 29 + 130r + 77r2 = 0 Begin completing the square. Divide all terms by 77 the coefficient of the squared term: Divide each side by '77'. 0.3766233766 + 1.688311688r + r2 = 0 Move the constant term to the right: Add '-0.3766233766' to each side of the equation. 0.3766233766 + 1.688311688r + -0.3766233766 + r2 = 0 + -0.3766233766 Reorder the terms: 0.3766233766 + -0.3766233766 + 1.688311688r + r2 = 0 + -0.3766233766 Combine like terms: 0.3766233766 + -0.3766233766 = 0.0000000000 0.0000000000 + 1.688311688r + r2 = 0 + -0.3766233766 1.688311688r + r2 = 0 + -0.3766233766 Combine like terms: 0 + -0.3766233766 = -0.3766233766 1.688311688r + r2 = -0.3766233766 The r term is 1.688311688r. Take half its coefficient (0.844155844). Square it (0.7125990890) and add it to both sides. Add '0.7125990890' to each side of the equation. 1.688311688r + 0.7125990890 + r2 = -0.3766233766 + 0.7125990890 Reorder the terms: 0.7125990890 + 1.688311688r + r2 = -0.3766233766 + 0.7125990890 Combine like terms: -0.3766233766 + 0.7125990890 = 0.3359757124 0.7125990890 + 1.688311688r + r2 = 0.3359757124 Factor a perfect square on the left side: (r + 0.844155844)(r + 0.844155844) = 0.3359757124 Calculate the square root of the right side: 0.579634119 Break this problem into two subproblems by setting (r + 0.844155844) equal to 0.579634119 and -0.579634119.Subproblem 1
r + 0.844155844 = 0.579634119 Simplifying r + 0.844155844 = 0.579634119 Reorder the terms: 0.844155844 + r = 0.579634119 Solving 0.844155844 + r = 0.579634119 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.844155844' to each side of the equation. 0.844155844 + -0.844155844 + r = 0.579634119 + -0.844155844 Combine like terms: 0.844155844 + -0.844155844 = 0.000000000 0.000000000 + r = 0.579634119 + -0.844155844 r = 0.579634119 + -0.844155844 Combine like terms: 0.579634119 + -0.844155844 = -0.264521725 r = -0.264521725 Simplifying r = -0.264521725Subproblem 2
r + 0.844155844 = -0.579634119 Simplifying r + 0.844155844 = -0.579634119 Reorder the terms: 0.844155844 + r = -0.579634119 Solving 0.844155844 + r = -0.579634119 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.844155844' to each side of the equation. 0.844155844 + -0.844155844 + r = -0.579634119 + -0.844155844 Combine like terms: 0.844155844 + -0.844155844 = 0.000000000 0.000000000 + r = -0.579634119 + -0.844155844 r = -0.579634119 + -0.844155844 Combine like terms: -0.579634119 + -0.844155844 = -1.423789963 r = -1.423789963 Simplifying r = -1.423789963Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.264521725, -1.423789963}
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