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Simplifying 4k2 = 16 + -1k Solving 4k2 = 16 + -1k Solving for variable 'k'. Reorder the terms: -16 + k + 4k2 = 16 + -1k + -16 + k Reorder the terms: -16 + k + 4k2 = 16 + -16 + -1k + k Combine like terms: 16 + -16 = 0 -16 + k + 4k2 = 0 + -1k + k -16 + k + 4k2 = -1k + k Combine like terms: -1k + k = 0 -16 + k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4 + 0.25k + k2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.25k + 4 + k2 = 0 + 4 Reorder the terms: -4 + 4 + 0.25k + k2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.25k + k2 = 0 + 4 0.25k + k2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.25k + k2 = 4 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25k + 0.25 + k2 = 4 + 0.25 Reorder the terms: 0.25 + 0.25k + k2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + 0.25k + k2 = 4.25 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (k + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
k + 0.5 = 2.061552813 Simplifying k + 0.5 = 2.061552813 Reorder the terms: 0.5 + k = 2.061552813 Solving 0.5 + k = 2.061552813 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 2.061552813 + -0.5 k = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 k = 1.561552813 Simplifying k = 1.561552813Subproblem 2
k + 0.5 = -2.061552813 Simplifying k + 0.5 = -2.061552813 Reorder the terms: 0.5 + k = -2.061552813 Solving 0.5 + k = -2.061552813 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -2.061552813 + -0.5 k = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 k = -2.561552813 Simplifying k = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.561552813, -2.561552813}
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