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Simplifying r2 + 3r + 0.5 = 0 Reorder the terms: 0.5 + 3r + r2 = 0 Solving 0.5 + 3r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 3r + -0.5 + r2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 3r + r2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 3r + r2 = 0 + -0.5 3r + r2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 3r + r2 = -0.5 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 = -0.5 + 2.25 Reorder the terms: 2.25 + 3r + r2 = -0.5 + 2.25 Combine like terms: -0.5 + 2.25 = 1.75 2.25 + 3r + r2 = 1.75 Factor a perfect square on the left side: (r + 1.5)(r + 1.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (r + 1.5) equal to 1.322875656 and -1.322875656.Subproblem 1
r + 1.5 = 1.322875656 Simplifying r + 1.5 = 1.322875656 Reorder the terms: 1.5 + r = 1.322875656 Solving 1.5 + r = 1.322875656 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 = 1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 1.322875656 + -1.5 r = 1.322875656 + -1.5 Combine like terms: 1.322875656 + -1.5 = -0.177124344 r = -0.177124344 Simplifying r = -0.177124344Subproblem 2
r + 1.5 = -1.322875656 Simplifying r + 1.5 = -1.322875656 Reorder the terms: 1.5 + r = -1.322875656 Solving 1.5 + r = -1.322875656 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 = -1.322875656 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -1.322875656 + -1.5 r = -1.322875656 + -1.5 Combine like terms: -1.322875656 + -1.5 = -2.822875656 r = -2.822875656 Simplifying r = -2.822875656Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.177124344, -2.822875656}
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