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Simplifying 3k2 + -10k + -18 = -6k2 + 3 Reorder the terms: -18 + -10k + 3k2 = -6k2 + 3 Reorder the terms: -18 + -10k + 3k2 = 3 + -6k2 Solving -18 + -10k + 3k2 = 3 + -6k2 Solving for variable 'k'. Reorder the terms: -18 + -3 + -10k + 3k2 + 6k2 = 3 + -6k2 + -3 + 6k2 Combine like terms: -18 + -3 = -21 -21 + -10k + 3k2 + 6k2 = 3 + -6k2 + -3 + 6k2 Combine like terms: 3k2 + 6k2 = 9k2 -21 + -10k + 9k2 = 3 + -6k2 + -3 + 6k2 Reorder the terms: -21 + -10k + 9k2 = 3 + -3 + -6k2 + 6k2 Combine like terms: 3 + -3 = 0 -21 + -10k + 9k2 = 0 + -6k2 + 6k2 -21 + -10k + 9k2 = -6k2 + 6k2 Combine like terms: -6k2 + 6k2 = 0 -21 + -10k + 9k2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.333333333 + -1.111111111k + k2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -1.111111111k + 2.333333333 + k2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -1.111111111k + k2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -1.111111111k + k2 = 0 + 2.333333333 -1.111111111k + k2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -1.111111111k + k2 = 2.333333333 The k term is -1.111111111k. Take half its coefficient (-0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. -1.111111111k + 0.3086419752 + k2 = 2.333333333 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111k + k2 = 2.333333333 + 0.3086419752 Combine like terms: 2.333333333 + 0.3086419752 = 2.6419753082 0.3086419752 + -1.111111111k + k2 = 2.6419753082 Factor a perfect square on the left side: (k + -0.5555555555)(k + -0.5555555555) = 2.6419753082 Calculate the square root of the right side: 1.625415426 Break this problem into two subproblems by setting (k + -0.5555555555) equal to 1.625415426 and -1.625415426.Subproblem 1
k + -0.5555555555 = 1.625415426 Simplifying k + -0.5555555555 = 1.625415426 Reorder the terms: -0.5555555555 + k = 1.625415426 Solving -0.5555555555 + k = 1.625415426 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + k = 1.625415426 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + k = 1.625415426 + 0.5555555555 k = 1.625415426 + 0.5555555555 Combine like terms: 1.625415426 + 0.5555555555 = 2.1809709815 k = 2.1809709815 Simplifying k = 2.1809709815Subproblem 2
k + -0.5555555555 = -1.625415426 Simplifying k + -0.5555555555 = -1.625415426 Reorder the terms: -0.5555555555 + k = -1.625415426 Solving -0.5555555555 + k = -1.625415426 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + k = -1.625415426 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + k = -1.625415426 + 0.5555555555 k = -1.625415426 + 0.5555555555 Combine like terms: -1.625415426 + 0.5555555555 = -1.0698598705 k = -1.0698598705 Simplifying k = -1.0698598705Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.1809709815, -1.0698598705}
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