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Simplifying 7k2 + -1(5k2 + -4k + 8) = 0 Reorder the terms: 7k2 + -1(8 + -4k + 5k2) = 0 7k2 + (8 * -1 + -4k * -1 + 5k2 * -1) = 0 7k2 + (-8 + 4k + -5k2) = 0 Reorder the terms: -8 + 4k + 7k2 + -5k2 = 0 Combine like terms: 7k2 + -5k2 = 2k2 -8 + 4k + 2k2 = 0 Solving -8 + 4k + 2k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 2k + k2) = 0 Ignore the factor 2.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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