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Simplifying (5k2 + 8k + 2) + -1(2k + 3) = 0 Reorder the terms: (2 + 8k + 5k2) + -1(2k + 3) = 0 Remove parenthesis around (2 + 8k + 5k2) 2 + 8k + 5k2 + -1(2k + 3) = 0 Reorder the terms: 2 + 8k + 5k2 + -1(3 + 2k) = 0 2 + 8k + 5k2 + (3 * -1 + 2k * -1) = 0 2 + 8k + 5k2 + (-3 + -2k) = 0 Reorder the terms: 2 + -3 + 8k + -2k + 5k2 = 0 Combine like terms: 2 + -3 = -1 -1 + 8k + -2k + 5k2 = 0 Combine like terms: 8k + -2k = 6k -1 + 6k + 5k2 = 0 Solving -1 + 6k + 5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + 1.2k + k2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 1.2k + 0.2 + k2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 1.2k + k2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 1.2k + k2 = 0 + 0.2 1.2k + k2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 1.2k + k2 = 0.2 The k term is 1.2k. Take half its coefficient (0.6). Square it (0.36) and add it to both sides. Add '0.36' to each side of the equation. 1.2k + 0.36 + k2 = 0.2 + 0.36 Reorder the terms: 0.36 + 1.2k + k2 = 0.2 + 0.36 Combine like terms: 0.2 + 0.36 = 0.56 0.36 + 1.2k + k2 = 0.56 Factor a perfect square on the left side: (k + 0.6)(k + 0.6) = 0.56 Calculate the square root of the right side: 0.748331477 Break this problem into two subproblems by setting (k + 0.6) equal to 0.748331477 and -0.748331477.Subproblem 1
k + 0.6 = 0.748331477 Simplifying k + 0.6 = 0.748331477 Reorder the terms: 0.6 + k = 0.748331477 Solving 0.6 + k = 0.748331477 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + k = 0.748331477 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + k = 0.748331477 + -0.6 k = 0.748331477 + -0.6 Combine like terms: 0.748331477 + -0.6 = 0.148331477 k = 0.148331477 Simplifying k = 0.148331477Subproblem 2
k + 0.6 = -0.748331477 Simplifying k + 0.6 = -0.748331477 Reorder the terms: 0.6 + k = -0.748331477 Solving 0.6 + k = -0.748331477 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + k = -0.748331477 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + k = -0.748331477 + -0.6 k = -0.748331477 + -0.6 Combine like terms: -0.748331477 + -0.6 = -1.348331477 k = -1.348331477 Simplifying k = -1.348331477Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.148331477, -1.348331477}
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