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Simplifying -16k2 = 20k + 24k + 5 Reorder the terms: -16k2 = 5 + 20k + 24k Combine like terms: 20k + 24k = 44k -16k2 = 5 + 44k Solving -16k2 = 5 + 44k Solving for variable 'k'. Reorder the terms: -5 + -44k + -16k2 = 5 + 44k + -5 + -44k Reorder the terms: -5 + -44k + -16k2 = 5 + -5 + 44k + -44k Combine like terms: 5 + -5 = 0 -5 + -44k + -16k2 = 0 + 44k + -44k -5 + -44k + -16k2 = 44k + -44k Combine like terms: 44k + -44k = 0 -5 + -44k + -16k2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 44k + 16k2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 44k + 16k2)' equal to zero and attempt to solve: Simplifying 5 + 44k + 16k2 = 0 Solving 5 + 44k + 16k2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + 2.75k + k2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + 2.75k + -0.3125 + k2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + 2.75k + k2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + 2.75k + k2 = 0 + -0.3125 2.75k + k2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 2.75k + k2 = -0.3125 The k term is 2.75k. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75k + 1.890625 + k2 = -0.3125 + 1.890625 Reorder the terms: 1.890625 + 2.75k + k2 = -0.3125 + 1.890625 Combine like terms: -0.3125 + 1.890625 = 1.578125 1.890625 + 2.75k + k2 = 1.578125 Factor a perfect square on the left side: (k + 1.375)(k + 1.375) = 1.578125 Calculate the square root of the right side: 1.256234453 Break this problem into two subproblems by setting (k + 1.375) equal to 1.256234453 and -1.256234453.Subproblem 1
k + 1.375 = 1.256234453 Simplifying k + 1.375 = 1.256234453 Reorder the terms: 1.375 + k = 1.256234453 Solving 1.375 + k = 1.256234453 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + k = 1.256234453 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + k = 1.256234453 + -1.375 k = 1.256234453 + -1.375 Combine like terms: 1.256234453 + -1.375 = -0.118765547 k = -0.118765547 Simplifying k = -0.118765547Subproblem 2
k + 1.375 = -1.256234453 Simplifying k + 1.375 = -1.256234453 Reorder the terms: 1.375 + k = -1.256234453 Solving 1.375 + k = -1.256234453 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + k = -1.256234453 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + k = -1.256234453 + -1.375 k = -1.256234453 + -1.375 Combine like terms: -1.256234453 + -1.375 = -2.631234453 k = -2.631234453 Simplifying k = -2.631234453Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.118765547, -2.631234453}Solution
k = {-0.118765547, -2.631234453}
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