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Simplifying 6k2 + -15 + 5k + -4k2 + -17k = 0 Reorder the terms: -15 + 5k + -17k + 6k2 + -4k2 = 0 Combine like terms: 5k + -17k = -12k -15 + -12k + 6k2 + -4k2 = 0 Combine like terms: 6k2 + -4k2 = 2k2 -15 + -12k + 2k2 = 0 Solving -15 + -12k + 2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + -6k + k2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -6k + 7.5 + k2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -6k + k2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -6k + k2 = 0 + 7.5 -6k + k2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -6k + k2 = 7.5 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 = 7.5 + 9 Reorder the terms: 9 + -6k + k2 = 7.5 + 9 Combine like terms: 7.5 + 9 = 16.5 9 + -6k + k2 = 16.5 Factor a perfect square on the left side: (k + -3)(k + -3) = 16.5 Calculate the square root of the right side: 4.062019202 Break this problem into two subproblems by setting (k + -3) equal to 4.062019202 and -4.062019202.Subproblem 1
k + -3 = 4.062019202 Simplifying k + -3 = 4.062019202 Reorder the terms: -3 + k = 4.062019202 Solving -3 + k = 4.062019202 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 = 4.062019202 + 3 Combine like terms: -3 + 3 = 0 0 + k = 4.062019202 + 3 k = 4.062019202 + 3 Combine like terms: 4.062019202 + 3 = 7.062019202 k = 7.062019202 Simplifying k = 7.062019202Subproblem 2
k + -3 = -4.062019202 Simplifying k + -3 = -4.062019202 Reorder the terms: -3 + k = -4.062019202 Solving -3 + k = -4.062019202 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 = -4.062019202 + 3 Combine like terms: -3 + 3 = 0 0 + k = -4.062019202 + 3 k = -4.062019202 + 3 Combine like terms: -4.062019202 + 3 = -1.062019202 k = -1.062019202 Simplifying k = -1.062019202Solution
The solution to the problem is based on the solutions from the subproblems. k = {7.062019202, -1.062019202}
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