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Simplifying (-4k4 + 14 + 3k2) + (3k4 + -14k4 + -8) = 0 Reorder the terms: (14 + 3k2 + -4k4) + (3k4 + -14k4 + -8) = 0 Remove parenthesis around (14 + 3k2 + -4k4) 14 + 3k2 + -4k4 + (3k4 + -14k4 + -8) = 0 Reorder the terms: 14 + 3k2 + -4k4 + (-8 + 3k4 + -14k4) = 0 Combine like terms: 3k4 + -14k4 = -11k4 14 + 3k2 + -4k4 + (-8 + -11k4) = 0 Remove parenthesis around (-8 + -11k4) 14 + 3k2 + -4k4 + -8 + -11k4 = 0 Reorder the terms: 14 + -8 + 3k2 + -4k4 + -11k4 = 0 Combine like terms: 14 + -8 = 6 6 + 3k2 + -4k4 + -11k4 = 0 Combine like terms: -4k4 + -11k4 = -15k4 6 + 3k2 + -15k4 = 0 Solving 6 + 3k2 + -15k4 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(2 + k2 + -5k4) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(2 + k2 + -5k4)' equal to zero and attempt to solve: Simplifying 2 + k2 + -5k4 = 0 Solving 2 + k2 + -5k4 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.4 + -0.2k2 + k4 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -0.2k2 + 0.4 + k4 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -0.2k2 + k4 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -0.2k2 + k4 = 0 + 0.4 -0.2k2 + k4 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -0.2k2 + k4 = 0.4 The k term is k2. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.2k2 + 0.25 + k4 = 0.4 + 0.25 Reorder the terms: 0.25 + -0.2k2 + k4 = 0.4 + 0.25 Combine like terms: 0.4 + 0.25 = 0.65 0.25 + -0.2k2 + k4 = 0.65 Factor a perfect square on the left side: (k2 + 0.5)(k2 + 0.5) = 0.65 Calculate the square root of the right side: 0.806225775 Break this problem into two subproblems by setting (k2 + 0.5) equal to 0.806225775 and -0.806225775.Subproblem 1
k2 + 0.5 = 0.806225775 Simplifying k2 + 0.5 = 0.806225775 Reorder the terms: 0.5 + k2 = 0.806225775 Solving 0.5 + k2 = 0.806225775 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k2 = 0.806225775 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k2 = 0.806225775 + -0.5 k2 = 0.806225775 + -0.5 Combine like terms: 0.806225775 + -0.5 = 0.306225775 k2 = 0.306225775 Simplifying k2 = 0.306225775 Take the square root of each side: k = {-0.553376703, 0.553376703}Subproblem 2
k2 + 0.5 = -0.806225775 Simplifying k2 + 0.5 = -0.806225775 Reorder the terms: 0.5 + k2 = -0.806225775 Solving 0.5 + k2 = -0.806225775 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k2 = -0.806225775 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k2 = -0.806225775 + -0.5 k2 = -0.806225775 + -0.5 Combine like terms: -0.806225775 + -0.5 = -1.306225775 k2 = -1.306225775 Simplifying k2 = -1.306225775 Reorder the terms: 1.306225775 + k2 = -1.306225775 + 1.306225775 Combine like terms: -1.306225775 + 1.306225775 = 0.000000000 1.306225775 + k2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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