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Simplifying 192r2 + 192r + -171 = 0 Reorder the terms: -171 + 192r + 192r2 = 0 Solving -171 + 192r + 192r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3'. 3(-57 + 64r + 64r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-57 + 64r + 64r2)' equal to zero and attempt to solve: Simplifying -57 + 64r + 64r2 = 0 Solving -57 + 64r + 64r2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -0.890625 + r + r2 = 0 Move the constant term to the right: Add '0.890625' to each side of the equation. -0.890625 + r + 0.890625 + r2 = 0 + 0.890625 Reorder the terms: -0.890625 + 0.890625 + r + r2 = 0 + 0.890625 Combine like terms: -0.890625 + 0.890625 = 0.000000 0.000000 + r + r2 = 0 + 0.890625 r + r2 = 0 + 0.890625 Combine like terms: 0 + 0.890625 = 0.890625 r + r2 = 0.890625 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 0.890625 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 0.890625 + 0.25 Combine like terms: 0.890625 + 0.25 = 1.140625 1.25 + r2 = 1.140625 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (r + 0.5) equal to 1.068000468 and -1.068000468.Subproblem 1
r + 0.5 = 1.068000468 Simplifying r + 0.5 = 1.068000468 Reorder the terms: 0.5 + r = 1.068000468 Solving 0.5 + r = 1.068000468 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 1.068000468 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.068000468 + -0.5 r = 1.068000468 + -0.5 Combine like terms: 1.068000468 + -0.5 = 0.568000468 r = 0.568000468 Simplifying r = 0.568000468Subproblem 2
r + 0.5 = -1.068000468 Simplifying r + 0.5 = -1.068000468 Reorder the terms: 0.5 + r = -1.068000468 Solving 0.5 + r = -1.068000468 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -1.068000468 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.068000468 + -0.5 r = -1.068000468 + -0.5 Combine like terms: -1.068000468 + -0.5 = -1.568000468 r = -1.568000468 Simplifying r = -1.568000468Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.568000468, -1.568000468}Solution
r = {0.568000468, -1.568000468}
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