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Simplifying 6k2 + 12k + -15 = 10 Reorder the terms: -15 + 12k + 6k2 = 10 Solving -15 + 12k + 6k2 = 10 Solving for variable 'k'. Reorder the terms: -15 + -10 + 12k + 6k2 = 10 + -10 Combine like terms: -15 + -10 = -25 -25 + 12k + 6k2 = 10 + -10 Combine like terms: 10 + -10 = 0 -25 + 12k + 6k2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.166666667 + 2k + k2 = 0 Move the constant term to the right: Add '4.166666667' to each side of the equation. -4.166666667 + 2k + 4.166666667 + k2 = 0 + 4.166666667 Reorder the terms: -4.166666667 + 4.166666667 + 2k + k2 = 0 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + 2k + k2 = 0 + 4.166666667 2k + k2 = 0 + 4.166666667 Combine like terms: 0 + 4.166666667 = 4.166666667 2k + k2 = 4.166666667 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 4.166666667 + 1 Reorder the terms: 1 + 2k + k2 = 4.166666667 + 1 Combine like terms: 4.166666667 + 1 = 5.166666667 1 + 2k + k2 = 5.166666667 Factor a perfect square on the left side: (k + 1)(k + 1) = 5.166666667 Calculate the square root of the right side: 2.273030283 Break this problem into two subproblems by setting (k + 1) equal to 2.273030283 and -2.273030283.Subproblem 1
k + 1 = 2.273030283 Simplifying k + 1 = 2.273030283 Reorder the terms: 1 + k = 2.273030283 Solving 1 + k = 2.273030283 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 2.273030283 + -1 Combine like terms: 1 + -1 = 0 0 + k = 2.273030283 + -1 k = 2.273030283 + -1 Combine like terms: 2.273030283 + -1 = 1.273030283 k = 1.273030283 Simplifying k = 1.273030283Subproblem 2
k + 1 = -2.273030283 Simplifying k + 1 = -2.273030283 Reorder the terms: 1 + k = -2.273030283 Solving 1 + k = -2.273030283 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -2.273030283 + -1 Combine like terms: 1 + -1 = 0 0 + k = -2.273030283 + -1 k = -2.273030283 + -1 Combine like terms: -2.273030283 + -1 = -3.273030283 k = -3.273030283 Simplifying k = -3.273030283Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.273030283, -3.273030283}
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