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Simplifying 16k4 + -40k2 + 13 = 0 Reorder the terms: 13 + -40k2 + 16k4 = 0 Solving 13 + -40k2 + 16k4 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.8125 + -2.5k2 + k4 = 0 Move the constant term to the right: Add '-0.8125' to each side of the equation. 0.8125 + -2.5k2 + -0.8125 + k4 = 0 + -0.8125 Reorder the terms: 0.8125 + -0.8125 + -2.5k2 + k4 = 0 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + -2.5k2 + k4 = 0 + -0.8125 -2.5k2 + k4 = 0 + -0.8125 Combine like terms: 0 + -0.8125 = -0.8125 -2.5k2 + k4 = -0.8125 The k term is -2.5k2. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5k2 + 1.5625 + k4 = -0.8125 + 1.5625 Reorder the terms: 1.5625 + -2.5k2 + k4 = -0.8125 + 1.5625 Combine like terms: -0.8125 + 1.5625 = 0.75 1.5625 + -2.5k2 + k4 = 0.75 Factor a perfect square on the left side: (k2 + -1.25)(k2 + -1.25) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (k2 + -1.25) equal to 0.866025404 and -0.866025404.Subproblem 1
k2 + -1.25 = 0.866025404 Simplifying k2 + -1.25 = 0.866025404 Reorder the terms: -1.25 + k2 = 0.866025404 Solving -1.25 + k2 = 0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + k2 = 0.866025404 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + k2 = 0.866025404 + 1.25 k2 = 0.866025404 + 1.25 Combine like terms: 0.866025404 + 1.25 = 2.116025404 k2 = 2.116025404 Simplifying k2 = 2.116025404 Take the square root of each side: k = {-1.454656456, 1.454656456}Subproblem 2
k2 + -1.25 = -0.866025404 Simplifying k2 + -1.25 = -0.866025404 Reorder the terms: -1.25 + k2 = -0.866025404 Solving -1.25 + k2 = -0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + k2 = -0.866025404 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + k2 = -0.866025404 + 1.25 k2 = -0.866025404 + 1.25 Combine like terms: -0.866025404 + 1.25 = 0.383974596 k2 = 0.383974596 Simplifying k2 = 0.383974596 Take the square root of each side: k = {-0.619656837, 0.619656837}Solution
The solution to the problem is based on the solutions from the subproblems. k = {-1.454656456, 1.454656456, -0.619656837, 0.619656837}
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