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Simplifying 32r2 + 82r + -16 = 0 Reorder the terms: -16 + 82r + 32r2 = 0 Solving -16 + 82r + 32r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-8 + 41r + 16r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-8 + 41r + 16r2)' equal to zero and attempt to solve: Simplifying -8 + 41r + 16r2 = 0 Solving -8 + 41r + 16r2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.5 + 2.5625r + r2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 2.5625r + 0.5 + r2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 2.5625r + r2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 2.5625r + r2 = 0 + 0.5 2.5625r + r2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 2.5625r + r2 = 0.5 The r term is 2.5625r. Take half its coefficient (1.28125). Square it (1.641601563) and add it to both sides. Add '1.641601563' to each side of the equation. 2.5625r + 1.641601563 + r2 = 0.5 + 1.641601563 Reorder the terms: 1.641601563 + 2.5625r + r2 = 0.5 + 1.641601563 Combine like terms: 0.5 + 1.641601563 = 2.141601563 1.641601563 + 2.5625r + r2 = 2.141601563 Factor a perfect square on the left side: (r + 1.28125)(r + 1.28125) = 2.141601563 Calculate the square root of the right side: 1.463421184 Break this problem into two subproblems by setting (r + 1.28125) equal to 1.463421184 and -1.463421184.Subproblem 1
r + 1.28125 = 1.463421184 Simplifying r + 1.28125 = 1.463421184 Reorder the terms: 1.28125 + r = 1.463421184 Solving 1.28125 + r = 1.463421184 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.28125' to each side of the equation. 1.28125 + -1.28125 + r = 1.463421184 + -1.28125 Combine like terms: 1.28125 + -1.28125 = 0.00000 0.00000 + r = 1.463421184 + -1.28125 r = 1.463421184 + -1.28125 Combine like terms: 1.463421184 + -1.28125 = 0.182171184 r = 0.182171184 Simplifying r = 0.182171184Subproblem 2
r + 1.28125 = -1.463421184 Simplifying r + 1.28125 = -1.463421184 Reorder the terms: 1.28125 + r = -1.463421184 Solving 1.28125 + r = -1.463421184 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.28125' to each side of the equation. 1.28125 + -1.28125 + r = -1.463421184 + -1.28125 Combine like terms: 1.28125 + -1.28125 = 0.00000 0.00000 + r = -1.463421184 + -1.28125 r = -1.463421184 + -1.28125 Combine like terms: -1.463421184 + -1.28125 = -2.744671184 r = -2.744671184 Simplifying r = -2.744671184Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.182171184, -2.744671184}Solution
r = {0.182171184, -2.744671184}
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