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Simplifying 3k2 + -12k + 10 = 0 Reorder the terms: 10 + -12k + 3k2 = 0 Solving 10 + -12k + 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'. 3.333333333 + -4k + k2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -4k + -3.333333333 + k2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -4k + k2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -4k + k2 = 0 + -3.333333333 -4k + k2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -4k + k2 = -3.333333333 The k term is -4k. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4k + 4 + k2 = -3.333333333 + 4 Reorder the terms: 4 + -4k + k2 = -3.333333333 + 4 Combine like terms: -3.333333333 + 4 = 0.666666667 4 + -4k + k2 = 0.666666667 Factor a perfect square on the left side: (k + -2)(k + -2) = 0.666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (k + -2) equal to 0.816496581 and -0.816496581.Subproblem 1
k + -2 = 0.816496581 Simplifying k + -2 = 0.816496581 Reorder the terms: -2 + k = 0.816496581 Solving -2 + k = 0.816496581 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + k = 0.816496581 + 2 Combine like terms: -2 + 2 = 0 0 + k = 0.816496581 + 2 k = 0.816496581 + 2 Combine like terms: 0.816496581 + 2 = 2.816496581 k = 2.816496581 Simplifying k = 2.816496581Subproblem 2
k + -2 = -0.816496581 Simplifying k + -2 = -0.816496581 Reorder the terms: -2 + k = -0.816496581 Solving -2 + k = -0.816496581 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + k = -0.816496581 + 2 Combine like terms: -2 + 2 = 0 0 + k = -0.816496581 + 2 k = -0.816496581 + 2 Combine like terms: -0.816496581 + 2 = 1.183503419 k = 1.183503419 Simplifying k = 1.183503419Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.816496581, 1.183503419}
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