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Simplifying 5r2 + -55r + 123 = 0 Reorder the terms: 123 + -55r + 5r2 = 0 Solving 123 + -55r + 5r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 24.6 + -11r + r2 = 0 Move the constant term to the right: Add '-24.6' to each side of the equation. 24.6 + -11r + -24.6 + r2 = 0 + -24.6 Reorder the terms: 24.6 + -24.6 + -11r + r2 = 0 + -24.6 Combine like terms: 24.6 + -24.6 = 0.0 0.0 + -11r + r2 = 0 + -24.6 -11r + r2 = 0 + -24.6 Combine like terms: 0 + -24.6 = -24.6 -11r + r2 = -24.6 The r term is -11r. Take half its coefficient (-5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. -11r + 30.25 + r2 = -24.6 + 30.25 Reorder the terms: 30.25 + -11r + r2 = -24.6 + 30.25 Combine like terms: -24.6 + 30.25 = 5.65 30.25 + -11r + r2 = 5.65 Factor a perfect square on the left side: (r + -5.5)(r + -5.5) = 5.65 Calculate the square root of the right side: 2.376972865 Break this problem into two subproblems by setting (r + -5.5) equal to 2.376972865 and -2.376972865.Subproblem 1
r + -5.5 = 2.376972865 Simplifying r + -5.5 = 2.376972865 Reorder the terms: -5.5 + r = 2.376972865 Solving -5.5 + r = 2.376972865 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + r = 2.376972865 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + r = 2.376972865 + 5.5 r = 2.376972865 + 5.5 Combine like terms: 2.376972865 + 5.5 = 7.876972865 r = 7.876972865 Simplifying r = 7.876972865Subproblem 2
r + -5.5 = -2.376972865 Simplifying r + -5.5 = -2.376972865 Reorder the terms: -5.5 + r = -2.376972865 Solving -5.5 + r = -2.376972865 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + r = -2.376972865 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + r = -2.376972865 + 5.5 r = -2.376972865 + 5.5 Combine like terms: -2.376972865 + 5.5 = 3.123027135 r = 3.123027135 Simplifying r = 3.123027135Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.876972865, 3.123027135}
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