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Simplifying -4k2 + -8k + -3 = 3 + -5k2 Reorder the terms: -3 + -8k + -4k2 = 3 + -5k2 Solving -3 + -8k + -4k2 = 3 + -5k2 Solving for variable 'k'. Reorder the terms: -3 + -3 + -8k + -4k2 + 5k2 = 3 + -5k2 + -3 + 5k2 Combine like terms: -3 + -3 = -6 -6 + -8k + -4k2 + 5k2 = 3 + -5k2 + -3 + 5k2 Combine like terms: -4k2 + 5k2 = 1k2 -6 + -8k + 1k2 = 3 + -5k2 + -3 + 5k2 Reorder the terms: -6 + -8k + 1k2 = 3 + -3 + -5k2 + 5k2 Combine like terms: 3 + -3 = 0 -6 + -8k + 1k2 = 0 + -5k2 + 5k2 -6 + -8k + 1k2 = -5k2 + 5k2 Combine like terms: -5k2 + 5k2 = 0 -6 + -8k + 1k2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -8k + 6 + k2 = 0 + 6 Reorder the terms: -6 + 6 + -8k + k2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -8k + k2 = 0 + 6 -8k + k2 = 0 + 6 Combine like terms: 0 + 6 = 6 -8k + k2 = 6 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 = 6 + 16 Reorder the terms: 16 + -8k + k2 = 6 + 16 Combine like terms: 6 + 16 = 22 16 + -8k + k2 = 22 Factor a perfect square on the left side: (k + -4)(k + -4) = 22 Calculate the square root of the right side: 4.69041576 Break this problem into two subproblems by setting (k + -4) equal to 4.69041576 and -4.69041576.Subproblem 1
k + -4 = 4.69041576 Simplifying k + -4 = 4.69041576 Reorder the terms: -4 + k = 4.69041576 Solving -4 + k = 4.69041576 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 = 4.69041576 + 4 Combine like terms: -4 + 4 = 0 0 + k = 4.69041576 + 4 k = 4.69041576 + 4 Combine like terms: 4.69041576 + 4 = 8.69041576 k = 8.69041576 Simplifying k = 8.69041576Subproblem 2
k + -4 = -4.69041576 Simplifying k + -4 = -4.69041576 Reorder the terms: -4 + k = -4.69041576 Solving -4 + k = -4.69041576 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 = -4.69041576 + 4 Combine like terms: -4 + 4 = 0 0 + k = -4.69041576 + 4 k = -4.69041576 + 4 Combine like terms: -4.69041576 + 4 = -0.69041576 k = -0.69041576 Simplifying k = -0.69041576Solution
The solution to the problem is based on the solutions from the subproblems. k = {8.69041576, -0.69041576}
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