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Simplifying 125 + r2 + 2r + 1 = 180 Reorder the terms: 125 + 1 + 2r + r2 = 180 Combine like terms: 125 + 1 = 126 126 + 2r + r2 = 180 Solving 126 + 2r + r2 = 180 Solving for variable 'r'. Reorder the terms: 126 + -180 + 2r + r2 = 180 + -180 Combine like terms: 126 + -180 = -54 -54 + 2r + r2 = 180 + -180 Combine like terms: 180 + -180 = 0 -54 + 2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '54' to each side of the equation. -54 + 2r + 54 + r2 = 0 + 54 Reorder the terms: -54 + 54 + 2r + r2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + 2r + r2 = 0 + 54 2r + r2 = 0 + 54 Combine like terms: 0 + 54 = 54 2r + r2 = 54 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 54 + 1 Reorder the terms: 1 + 2r + r2 = 54 + 1 Combine like terms: 54 + 1 = 55 1 + 2r + r2 = 55 Factor a perfect square on the left side: (r + 1)(r + 1) = 55 Calculate the square root of the right side: 7.416198487 Break this problem into two subproblems by setting (r + 1) equal to 7.416198487 and -7.416198487.Subproblem 1
r + 1 = 7.416198487 Simplifying r + 1 = 7.416198487 Reorder the terms: 1 + r = 7.416198487 Solving 1 + r = 7.416198487 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 7.416198487 + -1 Combine like terms: 1 + -1 = 0 0 + r = 7.416198487 + -1 r = 7.416198487 + -1 Combine like terms: 7.416198487 + -1 = 6.416198487 r = 6.416198487 Simplifying r = 6.416198487Subproblem 2
r + 1 = -7.416198487 Simplifying r + 1 = -7.416198487 Reorder the terms: 1 + r = -7.416198487 Solving 1 + r = -7.416198487 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -7.416198487 + -1 Combine like terms: 1 + -1 = 0 0 + r = -7.416198487 + -1 r = -7.416198487 + -1 Combine like terms: -7.416198487 + -1 = -8.416198487 r = -8.416198487 Simplifying r = -8.416198487Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.416198487, -8.416198487}
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