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Simplifying 4k2 = -8k + 16 Reorder the terms: 4k2 = 16 + -8k Solving 4k2 = 16 + -8k Solving for variable 'k'. Reorder the terms: -16 + 8k + 4k2 = 16 + -8k + -16 + 8k Reorder the terms: -16 + 8k + 4k2 = 16 + -16 + -8k + 8k Combine like terms: 16 + -16 = 0 -16 + 8k + 4k2 = 0 + -8k + 8k -16 + 8k + 4k2 = -8k + 8k Combine like terms: -8k + 8k = 0 -16 + 8k + 4k2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-4 + 2k + k2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-4 + 2k + k2)' equal to zero and attempt to solve: Simplifying -4 + 2k + k2 = 0 Solving -4 + 2k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 2k + 4 + k2 = 0 + 4 Reorder the terms: -4 + 4 + 2k + k2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2k + k2 = 0 + 4 2k + k2 = 0 + 4 Combine like terms: 0 + 4 = 4 2k + k2 = 4 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 4 + 1 Reorder the terms: 1 + 2k + k2 = 4 + 1 Combine like terms: 4 + 1 = 5 1 + 2k + k2 = 5 Factor a perfect square on the left side: (k + 1)(k + 1) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (k + 1) equal to 2.236067978 and -2.236067978.Subproblem 1
k + 1 = 2.236067978 Simplifying k + 1 = 2.236067978 Reorder the terms: 1 + k = 2.236067978 Solving 1 + k = 2.236067978 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + k = 2.236067978 + -1 k = 2.236067978 + -1 Combine like terms: 2.236067978 + -1 = 1.236067978 k = 1.236067978 Simplifying k = 1.236067978Subproblem 2
k + 1 = -2.236067978 Simplifying k + 1 = -2.236067978 Reorder the terms: 1 + k = -2.236067978 Solving 1 + k = -2.236067978 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -2.236067978 + -1 Combine like terms: 1 + -1 = 0 0 + k = -2.236067978 + -1 k = -2.236067978 + -1 Combine like terms: -2.236067978 + -1 = -3.236067978 k = -3.236067978 Simplifying k = -3.236067978Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.236067978, -3.236067978}Solution
k = {1.236067978, -3.236067978}
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