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Simplifying -51 + -23k = -10k2 + 1k Reorder the terms: -51 + -23k = 1k + -10k2 Solving -51 + -23k = 1k + -10k2 Solving for variable 'k'. Combine like terms: -23k + -1k = -24k -51 + -24k + 10k2 = 1k + -10k2 + -1k + 10k2 Reorder the terms: -51 + -24k + 10k2 = 1k + -1k + -10k2 + 10k2 Combine like terms: 1k + -1k = 0 -51 + -24k + 10k2 = 0 + -10k2 + 10k2 -51 + -24k + 10k2 = -10k2 + 10k2 Combine like terms: -10k2 + 10k2 = 0 -51 + -24k + 10k2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -5.1 + -2.4k + k2 = 0 Move the constant term to the right: Add '5.1' to each side of the equation. -5.1 + -2.4k + 5.1 + k2 = 0 + 5.1 Reorder the terms: -5.1 + 5.1 + -2.4k + k2 = 0 + 5.1 Combine like terms: -5.1 + 5.1 = 0.0 0.0 + -2.4k + k2 = 0 + 5.1 -2.4k + k2 = 0 + 5.1 Combine like terms: 0 + 5.1 = 5.1 -2.4k + k2 = 5.1 The k term is -2.4k. Take half its coefficient (-1.2). Square it (1.44) and add it to both sides. Add '1.44' to each side of the equation. -2.4k + 1.44 + k2 = 5.1 + 1.44 Reorder the terms: 1.44 + -2.4k + k2 = 5.1 + 1.44 Combine like terms: 5.1 + 1.44 = 6.54 1.44 + -2.4k + k2 = 6.54 Factor a perfect square on the left side: (k + -1.2)(k + -1.2) = 6.54 Calculate the square root of the right side: 2.557342371 Break this problem into two subproblems by setting (k + -1.2) equal to 2.557342371 and -2.557342371.Subproblem 1
k + -1.2 = 2.557342371 Simplifying k + -1.2 = 2.557342371 Reorder the terms: -1.2 + k = 2.557342371 Solving -1.2 + k = 2.557342371 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + k = 2.557342371 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + k = 2.557342371 + 1.2 k = 2.557342371 + 1.2 Combine like terms: 2.557342371 + 1.2 = 3.757342371 k = 3.757342371 Simplifying k = 3.757342371Subproblem 2
k + -1.2 = -2.557342371 Simplifying k + -1.2 = -2.557342371 Reorder the terms: -1.2 + k = -2.557342371 Solving -1.2 + k = -2.557342371 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + k = -2.557342371 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + k = -2.557342371 + 1.2 k = -2.557342371 + 1.2 Combine like terms: -2.557342371 + 1.2 = -1.357342371 k = -1.357342371 Simplifying k = -1.357342371Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.757342371, -1.357342371}
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