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Simplifying -1k + -4 + -4k + -3 = -1 + -3k(k) Reorder the terms: -4 + -3 + -1k + -4k = -1 + -3k(k) Combine like terms: -4 + -3 = -7 -7 + -1k + -4k = -1 + -3k(k) Combine like terms: -1k + -4k = -5k -7 + -5k = -1 + -3k(k) Multiply k * k -7 + -5k = -1 + -3k2 Solving -7 + -5k = -1 + -3k2 Solving for variable 'k'. Reorder the terms: -7 + 1 + -5k + 3k2 = -1 + -3k2 + 1 + 3k2 Combine like terms: -7 + 1 = -6 -6 + -5k + 3k2 = -1 + -3k2 + 1 + 3k2 Reorder the terms: -6 + -5k + 3k2 = -1 + 1 + -3k2 + 3k2 Combine like terms: -1 + 1 = 0 -6 + -5k + 3k2 = 0 + -3k2 + 3k2 -6 + -5k + 3k2 = -3k2 + 3k2 Combine like terms: -3k2 + 3k2 = 0 -6 + -5k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + -1.666666667k + k2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.666666667k + 2 + k2 = 0 + 2 Reorder the terms: -2 + 2 + -1.666666667k + k2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.666666667k + k2 = 0 + 2 -1.666666667k + k2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.666666667k + k2 = 2 The k term is -1.666666667k. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667k + 0.6944444447 + k2 = 2 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667k + k2 = 2 + 0.6944444447 Combine like terms: 2 + 0.6944444447 = 2.6944444447 0.6944444447 + -1.666666667k + k2 = 2.6944444447 Factor a perfect square on the left side: (k + -0.8333333335)(k + -0.8333333335) = 2.6944444447 Calculate the square root of the right side: 1.6414763 Break this problem into two subproblems by setting (k + -0.8333333335) equal to 1.6414763 and -1.6414763.Subproblem 1
k + -0.8333333335 = 1.6414763 Simplifying k + -0.8333333335 = 1.6414763 Reorder the terms: -0.8333333335 + k = 1.6414763 Solving -0.8333333335 + k = 1.6414763 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + k = 1.6414763 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + k = 1.6414763 + 0.8333333335 k = 1.6414763 + 0.8333333335 Combine like terms: 1.6414763 + 0.8333333335 = 2.4748096335 k = 2.4748096335 Simplifying k = 2.4748096335Subproblem 2
k + -0.8333333335 = -1.6414763 Simplifying k + -0.8333333335 = -1.6414763 Reorder the terms: -0.8333333335 + k = -1.6414763 Solving -0.8333333335 + k = -1.6414763 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + k = -1.6414763 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + k = -1.6414763 + 0.8333333335 k = -1.6414763 + 0.8333333335 Combine like terms: -1.6414763 + 0.8333333335 = -0.8081429665 k = -0.8081429665 Simplifying k = -0.8081429665Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.4748096335, -0.8081429665}
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