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Simplifying (8k + k2 + -6) + -1(-10k + 7 + -2k2) = 0 Reorder the terms: (-6 + 8k + k2) + -1(-10k + 7 + -2k2) = 0 Remove parenthesis around (-6 + 8k + k2) -6 + 8k + k2 + -1(-10k + 7 + -2k2) = 0 Reorder the terms: -6 + 8k + k2 + -1(7 + -10k + -2k2) = 0 -6 + 8k + k2 + (7 * -1 + -10k * -1 + -2k2 * -1) = 0 -6 + 8k + k2 + (-7 + 10k + 2k2) = 0 Reorder the terms: -6 + -7 + 8k + 10k + k2 + 2k2 = 0 Combine like terms: -6 + -7 = -13 -13 + 8k + 10k + k2 + 2k2 = 0 Combine like terms: 8k + 10k = 18k -13 + 18k + k2 + 2k2 = 0 Combine like terms: k2 + 2k2 = 3k2 -13 + 18k + 3k2 = 0 Solving -13 + 18k + 3k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4.333333333 + 6k + k2 = 0 Move the constant term to the right: Add '4.333333333' to each side of the equation. -4.333333333 + 6k + 4.333333333 + k2 = 0 + 4.333333333 Reorder the terms: -4.333333333 + 4.333333333 + 6k + k2 = 0 + 4.333333333 Combine like terms: -4.333333333 + 4.333333333 = 0.000000000 0.000000000 + 6k + k2 = 0 + 4.333333333 6k + k2 = 0 + 4.333333333 Combine like terms: 0 + 4.333333333 = 4.333333333 6k + k2 = 4.333333333 The k term is 6k. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6k + 9 + k2 = 4.333333333 + 9 Reorder the terms: 9 + 6k + k2 = 4.333333333 + 9 Combine like terms: 4.333333333 + 9 = 13.333333333 9 + 6k + k2 = 13.333333333 Factor a perfect square on the left side: (k + 3)(k + 3) = 13.333333333 Calculate the square root of the right side: 3.651483717 Break this problem into two subproblems by setting (k + 3) equal to 3.651483717 and -3.651483717.Subproblem 1
k + 3 = 3.651483717 Simplifying k + 3 = 3.651483717 Reorder the terms: 3 + k = 3.651483717 Solving 3 + k = 3.651483717 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = 3.651483717 + -3 Combine like terms: 3 + -3 = 0 0 + k = 3.651483717 + -3 k = 3.651483717 + -3 Combine like terms: 3.651483717 + -3 = 0.651483717 k = 0.651483717 Simplifying k = 0.651483717Subproblem 2
k + 3 = -3.651483717 Simplifying k + 3 = -3.651483717 Reorder the terms: 3 + k = -3.651483717 Solving 3 + k = -3.651483717 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = -3.651483717 + -3 Combine like terms: 3 + -3 = 0 0 + k = -3.651483717 + -3 k = -3.651483717 + -3 Combine like terms: -3.651483717 + -3 = -6.651483717 k = -6.651483717 Simplifying k = -6.651483717Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.651483717, -6.651483717}
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