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Simplifying k2 + 3k + -90 = 0 Reorder the terms: -90 + 3k + k2 = 0 Solving -90 + 3k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '90' to each side of the equation. -90 + 3k + 90 + k2 = 0 + 90 Reorder the terms: -90 + 90 + 3k + k2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 3k + k2 = 0 + 90 3k + k2 = 0 + 90 Combine like terms: 0 + 90 = 90 3k + k2 = 90 The k term is 3k. 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. 3k + 2.25 + k2 = 90 + 2.25 Reorder the terms: 2.25 + 3k + k2 = 90 + 2.25 Combine like terms: 90 + 2.25 = 92.25 2.25 + 3k + k2 = 92.25 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 92.25 Calculate the square root of the right side: 9.604686356 Break this problem into two subproblems by setting (k + 1.5) equal to 9.604686356 and -9.604686356.Subproblem 1
k + 1.5 = 9.604686356 Simplifying k + 1.5 = 9.604686356 Reorder the terms: 1.5 + k = 9.604686356 Solving 1.5 + k = 9.604686356 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 9.604686356 + -1.5 k = 9.604686356 + -1.5 Combine like terms: 9.604686356 + -1.5 = 8.104686356 k = 8.104686356 Simplifying k = 8.104686356Subproblem 2
k + 1.5 = -9.604686356 Simplifying k + 1.5 = -9.604686356 Reorder the terms: 1.5 + k = -9.604686356 Solving 1.5 + k = -9.604686356 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -9.604686356 + -1.5 k = -9.604686356 + -1.5 Combine like terms: -9.604686356 + -1.5 = -11.104686356 k = -11.104686356 Simplifying k = -11.104686356Solution
The solution to the problem is based on the solutions from the subproblems. k = {8.104686356, -11.104686356}
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