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Simplifying (8 + -8k2 + -2k) + -1(4 + -4k2 + -7k) = 0 Reorder the terms: (8 + -2k + -8k2) + -1(4 + -4k2 + -7k) = 0 Remove parenthesis around (8 + -2k + -8k2) 8 + -2k + -8k2 + -1(4 + -4k2 + -7k) = 0 Reorder the terms: 8 + -2k + -8k2 + -1(4 + -7k + -4k2) = 0 8 + -2k + -8k2 + (4 * -1 + -7k * -1 + -4k2 * -1) = 0 8 + -2k + -8k2 + (-4 + 7k + 4k2) = 0 Reorder the terms: 8 + -4 + -2k + 7k + -8k2 + 4k2 = 0 Combine like terms: 8 + -4 = 4 4 + -2k + 7k + -8k2 + 4k2 = 0 Combine like terms: -2k + 7k = 5k 4 + 5k + -8k2 + 4k2 = 0 Combine like terms: -8k2 + 4k2 = -4k2 4 + 5k + -4k2 = 0 Solving 4 + 5k + -4k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1 + -1.25k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.25k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + -1.25k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.25k + k2 = 0 + 1 -1.25k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1.25k + k2 = 1 The k term is -1.25k. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25k + 0.390625 + k2 = 1 + 0.390625 Reorder the terms: 0.390625 + -1.25k + k2 = 1 + 0.390625 Combine like terms: 1 + 0.390625 = 1.390625 0.390625 + -1.25k + k2 = 1.390625 Factor a perfect square on the left side: (k + -0.625)(k + -0.625) = 1.390625 Calculate the square root of the right side: 1.179247642 Break this problem into two subproblems by setting (k + -0.625) equal to 1.179247642 and -1.179247642.Subproblem 1
k + -0.625 = 1.179247642 Simplifying k + -0.625 = 1.179247642 Reorder the terms: -0.625 + k = 1.179247642 Solving -0.625 + k = 1.179247642 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = 1.179247642 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = 1.179247642 + 0.625 k = 1.179247642 + 0.625 Combine like terms: 1.179247642 + 0.625 = 1.804247642 k = 1.804247642 Simplifying k = 1.804247642Subproblem 2
k + -0.625 = -1.179247642 Simplifying k + -0.625 = -1.179247642 Reorder the terms: -0.625 + k = -1.179247642 Solving -0.625 + k = -1.179247642 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + k = -1.179247642 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + k = -1.179247642 + 0.625 k = -1.179247642 + 0.625 Combine like terms: -1.179247642 + 0.625 = -0.554247642 k = -0.554247642 Simplifying k = -0.554247642Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.804247642, -0.554247642}
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