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Simplifying r2 + 11r + -78 = 0 Reorder the terms: -78 + 11r + r2 = 0 Solving -78 + 11r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 11r + 78 + r2 = 0 + 78 Reorder the terms: -78 + 78 + 11r + r2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 11r + r2 = 0 + 78 11r + r2 = 0 + 78 Combine like terms: 0 + 78 = 78 11r + r2 = 78 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 = 78 + 30.25 Reorder the terms: 30.25 + 11r + r2 = 78 + 30.25 Combine like terms: 78 + 30.25 = 108.25 30.25 + 11r + r2 = 108.25 Factor a perfect square on the left side: (r + 5.5)(r + 5.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (r + 5.5) equal to 10.404326023 and -10.404326023.Subproblem 1
r + 5.5 = 10.404326023 Simplifying r + 5.5 = 10.404326023 Reorder the terms: 5.5 + r = 10.404326023 Solving 5.5 + r = 10.404326023 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 = 10.404326023 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + r = 10.404326023 + -5.5 r = 10.404326023 + -5.5 Combine like terms: 10.404326023 + -5.5 = 4.904326023 r = 4.904326023 Simplifying r = 4.904326023Subproblem 2
r + 5.5 = -10.404326023 Simplifying r + 5.5 = -10.404326023 Reorder the terms: 5.5 + r = -10.404326023 Solving 5.5 + r = -10.404326023 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 = -10.404326023 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + r = -10.404326023 + -5.5 r = -10.404326023 + -5.5 Combine like terms: -10.404326023 + -5.5 = -15.904326023 r = -15.904326023 Simplifying r = -15.904326023Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.904326023, -15.904326023}
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