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Simplifying 4k2 + 28k + 48 = 2 Reorder the terms: 48 + 28k + 4k2 = 2 Solving 48 + 28k + 4k2 = 2 Solving for variable 'k'. Reorder the terms: 48 + -2 + 28k + 4k2 = 2 + -2 Combine like terms: 48 + -2 = 46 46 + 28k + 4k2 = 2 + -2 Combine like terms: 2 + -2 = 0 46 + 28k + 4k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(23 + 14k + 2k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(23 + 14k + 2k2)' equal to zero and attempt to solve: Simplifying 23 + 14k + 2k2 = 0 Solving 23 + 14k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 11.5 + 7k + k2 = 0 Move the constant term to the right: Add '-11.5' to each side of the equation. 11.5 + 7k + -11.5 + k2 = 0 + -11.5 Reorder the terms: 11.5 + -11.5 + 7k + k2 = 0 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + 7k + k2 = 0 + -11.5 7k + k2 = 0 + -11.5 Combine like terms: 0 + -11.5 = -11.5 7k + k2 = -11.5 The k term is 7k. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7k + 12.25 + k2 = -11.5 + 12.25 Reorder the terms: 12.25 + 7k + k2 = -11.5 + 12.25 Combine like terms: -11.5 + 12.25 = 0.75 12.25 + 7k + k2 = 0.75 Factor a perfect square on the left side: (k + 3.5)(k + 3.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (k + 3.5) equal to 0.866025404 and -0.866025404.Subproblem 1
k + 3.5 = 0.866025404 Simplifying k + 3.5 = 0.866025404 Reorder the terms: 3.5 + k = 0.866025404 Solving 3.5 + k = 0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = 0.866025404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = 0.866025404 + -3.5 k = 0.866025404 + -3.5 Combine like terms: 0.866025404 + -3.5 = -2.633974596 k = -2.633974596 Simplifying k = -2.633974596Subproblem 2
k + 3.5 = -0.866025404 Simplifying k + 3.5 = -0.866025404 Reorder the terms: 3.5 + k = -0.866025404 Solving 3.5 + k = -0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = -0.866025404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = -0.866025404 + -3.5 k = -0.866025404 + -3.5 Combine like terms: -0.866025404 + -3.5 = -4.366025404 k = -4.366025404 Simplifying k = -4.366025404Solution
The solution to the problem is based on the solutions from the subproblems. k = {-2.633974596, -4.366025404}Solution
k = {-2.633974596, -4.366025404}
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