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Simplifying r2 + 5r + 3 = 0 Reorder the terms: 3 + 5r + r2 = 0 Solving 3 + 5r + r2 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + 5r + -3 + r2 = 0 + -3 Reorder the terms: 3 + -3 + 5r + r2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 5r + r2 = 0 + -3 5r + r2 = 0 + -3 Combine like terms: 0 + -3 = -3 5r + r2 = -3 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = -3 + 6.25 Reorder the terms: 6.25 + 5r + r2 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + 5r + r2 = 3.25 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (r + 2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
r + 2.5 = 1.802775638 Simplifying r + 2.5 = 1.802775638 Reorder the terms: 2.5 + r = 1.802775638 Solving 2.5 + r = 1.802775638 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 1.802775638 + -2.5 r = 1.802775638 + -2.5 Combine like terms: 1.802775638 + -2.5 = -0.697224362 r = -0.697224362 Simplifying r = -0.697224362Subproblem 2
r + 2.5 = -1.802775638 Simplifying r + 2.5 = -1.802775638 Reorder the terms: 2.5 + r = -1.802775638 Solving 2.5 + r = -1.802775638 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -1.802775638 + -2.5 r = -1.802775638 + -2.5 Combine like terms: -1.802775638 + -2.5 = -4.302775638 r = -4.302775638 Simplifying r = -4.302775638Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.697224362, -4.302775638}
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