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Simplifying 2k2 + 8k + -8 = 0 Reorder the terms: -8 + 8k + 2k2 = 0 Solving -8 + 8k + 2k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 4k + k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + 4k + k2)' equal to zero and attempt to solve: Simplifying -4 + 4k + k2 = 0 Solving -4 + 4k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 4k + 4 + k2 = 0 + 4 Reorder the terms: -4 + 4 + 4k + k2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 4k + k2 = 0 + 4 4k + k2 = 0 + 4 Combine like terms: 0 + 4 = 4 4k + k2 = 4 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 = 4 + 4 Reorder the terms: 4 + 4k + k2 = 4 + 4 Combine like terms: 4 + 4 = 8 4 + 4k + k2 = 8 Factor a perfect square on the left side: (k + 2)(k + 2) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (k + 2) equal to 2.828427125 and -2.828427125.Subproblem 1
k + 2 = 2.828427125 Simplifying k + 2 = 2.828427125 Reorder the terms: 2 + k = 2.828427125 Solving 2 + k = 2.828427125 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 = 2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + k = 2.828427125 + -2 k = 2.828427125 + -2 Combine like terms: 2.828427125 + -2 = 0.828427125 k = 0.828427125 Simplifying k = 0.828427125Subproblem 2
k + 2 = -2.828427125 Simplifying k + 2 = -2.828427125 Reorder the terms: 2 + k = -2.828427125 Solving 2 + k = -2.828427125 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 = -2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + k = -2.828427125 + -2 k = -2.828427125 + -2 Combine like terms: -2.828427125 + -2 = -4.828427125 k = -4.828427125 Simplifying k = -4.828427125Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.828427125, -4.828427125}Solution
k = {0.828427125, -4.828427125}
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