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Simplifying k2 + -16k + 64 = 8 Reorder the terms: 64 + -16k + k2 = 8 Solving 64 + -16k + k2 = 8 Solving for variable 'k'. Reorder the terms: 64 + -8 + -16k + k2 = 8 + -8 Combine like terms: 64 + -8 = 56 56 + -16k + k2 = 8 + -8 Combine like terms: 8 + -8 = 0 56 + -16k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-56' to each side of the equation. 56 + -16k + -56 + k2 = 0 + -56 Reorder the terms: 56 + -56 + -16k + k2 = 0 + -56 Combine like terms: 56 + -56 = 0 0 + -16k + k2 = 0 + -56 -16k + k2 = 0 + -56 Combine like terms: 0 + -56 = -56 -16k + k2 = -56 The k term is -16k. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16k + 64 + k2 = -56 + 64 Reorder the terms: 64 + -16k + k2 = -56 + 64 Combine like terms: -56 + 64 = 8 64 + -16k + k2 = 8 Factor a perfect square on the left side: (k + -8)(k + -8) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (k + -8) equal to 2.828427125 and -2.828427125.Subproblem 1
k + -8 = 2.828427125 Simplifying k + -8 = 2.828427125 Reorder the terms: -8 + k = 2.828427125 Solving -8 + k = 2.828427125 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + k = 2.828427125 + 8 Combine like terms: -8 + 8 = 0 0 + k = 2.828427125 + 8 k = 2.828427125 + 8 Combine like terms: 2.828427125 + 8 = 10.828427125 k = 10.828427125 Simplifying k = 10.828427125Subproblem 2
k + -8 = -2.828427125 Simplifying k + -8 = -2.828427125 Reorder the terms: -8 + k = -2.828427125 Solving -8 + k = -2.828427125 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + k = -2.828427125 + 8 Combine like terms: -8 + 8 = 0 0 + k = -2.828427125 + 8 k = -2.828427125 + 8 Combine like terms: -2.828427125 + 8 = 5.171572875 k = 5.171572875 Simplifying k = 5.171572875Solution
The solution to the problem is based on the solutions from the subproblems. k = {10.828427125, 5.171572875}
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