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Simplifying k(k + 1) + -549 = 0 Reorder the terms: k(1 + k) + -549 = 0 (1 * k + k * k) + -549 = 0 (1k + k2) + -549 = 0 Reorder the terms: -549 + 1k + k2 = 0 Solving -549 + 1k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '549' to each side of the equation. -549 + 1k + 549 + k2 = 0 + 549 Reorder the terms: -549 + 549 + 1k + k2 = 0 + 549 Combine like terms: -549 + 549 = 0 0 + 1k + k2 = 0 + 549 1k + k2 = 0 + 549 Combine like terms: 0 + 549 = 549 1k + k2 = 549 The k term is 1k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. k + 0.25 + k2 = 549 + 0.25 Reorder the terms: 0.25 + k + k2 = 549 + 0.25 Combine like terms: 549 + 0.25 = 549.25 0.25 + k + k2 = 549.25 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 549.25 Calculate the square root of the right side: 23.436083291 Break this problem into two subproblems by setting (k + 0.5) equal to 23.436083291 and -23.436083291.Subproblem 1
k + 0.5 = 23.436083291 Simplifying k + 0.5 = 23.436083291 Reorder the terms: 0.5 + k = 23.436083291 Solving 0.5 + k = 23.436083291 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 23.436083291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 23.436083291 + -0.5 k = 23.436083291 + -0.5 Combine like terms: 23.436083291 + -0.5 = 22.936083291 k = 22.936083291 Simplifying k = 22.936083291Subproblem 2
k + 0.5 = -23.436083291 Simplifying k + 0.5 = -23.436083291 Reorder the terms: 0.5 + k = -23.436083291 Solving 0.5 + k = -23.436083291 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -23.436083291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -23.436083291 + -0.5 k = -23.436083291 + -0.5 Combine like terms: -23.436083291 + -0.5 = -23.936083291 k = -23.936083291 Simplifying k = -23.936083291Solution
The solution to the problem is based on the solutions from the subproblems. k = {22.936083291, -23.936083291}
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