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Simplifying k2 + 14k + -78 = -10 Reorder the terms: -78 + 14k + k2 = -10 Solving -78 + 14k + k2 = -10 Solving for variable 'k'. Reorder the terms: -78 + 10 + 14k + k2 = -10 + 10 Combine like terms: -78 + 10 = -68 -68 + 14k + k2 = -10 + 10 Combine like terms: -10 + 10 = 0 -68 + 14k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '68' to each side of the equation. -68 + 14k + 68 + k2 = 0 + 68 Reorder the terms: -68 + 68 + 14k + k2 = 0 + 68 Combine like terms: -68 + 68 = 0 0 + 14k + k2 = 0 + 68 14k + k2 = 0 + 68 Combine like terms: 0 + 68 = 68 14k + k2 = 68 The k term is 14k. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14k + 49 + k2 = 68 + 49 Reorder the terms: 49 + 14k + k2 = 68 + 49 Combine like terms: 68 + 49 = 117 49 + 14k + k2 = 117 Factor a perfect square on the left side: (k + 7)(k + 7) = 117 Calculate the square root of the right side: 10.816653826 Break this problem into two subproblems by setting (k + 7) equal to 10.816653826 and -10.816653826.Subproblem 1
k + 7 = 10.816653826 Simplifying k + 7 = 10.816653826 Reorder the terms: 7 + k = 10.816653826 Solving 7 + k = 10.816653826 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + k = 10.816653826 + -7 Combine like terms: 7 + -7 = 0 0 + k = 10.816653826 + -7 k = 10.816653826 + -7 Combine like terms: 10.816653826 + -7 = 3.816653826 k = 3.816653826 Simplifying k = 3.816653826Subproblem 2
k + 7 = -10.816653826 Simplifying k + 7 = -10.816653826 Reorder the terms: 7 + k = -10.816653826 Solving 7 + k = -10.816653826 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + k = -10.816653826 + -7 Combine like terms: 7 + -7 = 0 0 + k = -10.816653826 + -7 k = -10.816653826 + -7 Combine like terms: -10.816653826 + -7 = -17.816653826 k = -17.816653826 Simplifying k = -17.816653826Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.816653826, -17.816653826}
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