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Simplifying r2 + 3r + -5 = 11 Reorder the terms: -5 + 3r + r2 = 11 Solving -5 + 3r + r2 = 11 Solving for variable 'r'. Reorder the terms: -5 + -11 + 3r + r2 = 11 + -11 Combine like terms: -5 + -11 = -16 -16 + 3r + r2 = 11 + -11 Combine like terms: 11 + -11 = 0 -16 + 3r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '16' to each side of the equation. -16 + 3r + 16 + r2 = 0 + 16 Reorder the terms: -16 + 16 + 3r + r2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 3r + r2 = 0 + 16 3r + r2 = 0 + 16 Combine like terms: 0 + 16 = 16 3r + r2 = 16 The r term is 3r. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3r + 2.25 + r2 = 16 + 2.25 Reorder the terms: 2.25 + 3r + r2 = 16 + 2.25 Combine like terms: 16 + 2.25 = 18.25 2.25 + 3r + r2 = 18.25 Factor a perfect square on the left side: (r + 1.5)(r + 1.5) = 18.25 Calculate the square root of the right side: 4.272001873 Break this problem into two subproblems by setting (r + 1.5) equal to 4.272001873 and -4.272001873.Subproblem 1
r + 1.5 = 4.272001873 Simplifying r + 1.5 = 4.272001873 Reorder the terms: 1.5 + r = 4.272001873 Solving 1.5 + r = 4.272001873 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = 4.272001873 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 4.272001873 + -1.5 r = 4.272001873 + -1.5 Combine like terms: 4.272001873 + -1.5 = 2.772001873 r = 2.772001873 Simplifying r = 2.772001873Subproblem 2
r + 1.5 = -4.272001873 Simplifying r + 1.5 = -4.272001873 Reorder the terms: 1.5 + r = -4.272001873 Solving 1.5 + r = -4.272001873 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + r = -4.272001873 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -4.272001873 + -1.5 r = -4.272001873 + -1.5 Combine like terms: -4.272001873 + -1.5 = -5.772001873 r = -5.772001873 Simplifying r = -5.772001873Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.772001873, -5.772001873}
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