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Simplifying -4k2 + -8k = -3 + -5k2 Reorder the terms: -8k + -4k2 = -3 + -5k2 Solving -8k + -4k2 = -3 + -5k2 Solving for variable 'k'. Reorder the terms: 3 + -8k + -4k2 + 5k2 = -3 + -5k2 + 3 + 5k2 Combine like terms: -4k2 + 5k2 = 1k2 3 + -8k + 1k2 = -3 + -5k2 + 3 + 5k2 Reorder the terms: 3 + -8k + 1k2 = -3 + 3 + -5k2 + 5k2 Combine like terms: -3 + 3 = 0 3 + -8k + 1k2 = 0 + -5k2 + 5k2 3 + -8k + 1k2 = -5k2 + 5k2 Combine like terms: -5k2 + 5k2 = 0 3 + -8k + 1k2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -8k + -3 + k2 = 0 + -3 Reorder the terms: 3 + -3 + -8k + k2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -8k + k2 = 0 + -3 -8k + k2 = 0 + -3 Combine like terms: 0 + -3 = -3 -8k + k2 = -3 The k term is -8k. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8k + 16 + k2 = -3 + 16 Reorder the terms: 16 + -8k + k2 = -3 + 16 Combine like terms: -3 + 16 = 13 16 + -8k + k2 = 13 Factor a perfect square on the left side: (k + -4)(k + -4) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (k + -4) equal to 3.605551275 and -3.605551275.Subproblem 1
k + -4 = 3.605551275 Simplifying k + -4 = 3.605551275 Reorder the terms: -4 + k = 3.605551275 Solving -4 + k = 3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = 3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + k = 3.605551275 + 4 k = 3.605551275 + 4 Combine like terms: 3.605551275 + 4 = 7.605551275 k = 7.605551275 Simplifying k = 7.605551275Subproblem 2
k + -4 = -3.605551275 Simplifying k + -4 = -3.605551275 Reorder the terms: -4 + k = -3.605551275 Solving -4 + k = -3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + k = -3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + k = -3.605551275 + 4 k = -3.605551275 + 4 Combine like terms: -3.605551275 + 4 = 0.394448725 k = 0.394448725 Simplifying k = 0.394448725Solution
The solution to the problem is based on the solutions from the subproblems. k = {7.605551275, 0.394448725}
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