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Simplifying 10 = k(4k + -1) Reorder the terms: 10 = k(-1 + 4k) 10 = (-1 * k + 4k * k) 10 = (-1k + 4k2) Solving 10 = -1k + 4k2 Solving for variable 'k'. Reorder the terms: 10 + k + -4k2 = -1k + k + 4k2 + -4k2 Combine like terms: -1k + k = 0 10 + k + -4k2 = 0 + 4k2 + -4k2 10 + k + -4k2 = 4k2 + -4k2 Combine like terms: 4k2 + -4k2 = 0 10 + k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.5 + -0.25k + k2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -0.25k + 2.5 + k2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -0.25k + k2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -0.25k + k2 = 0 + 2.5 -0.25k + k2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -0.25k + k2 = 2.5 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 = 2.5 + 0.25 Reorder the terms: 0.25 + -0.25k + k2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 0.25 + -0.25k + k2 = 2.75 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (k + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
k + 0.5 = 1.658312395 Simplifying k + 0.5 = 1.658312395 Reorder the terms: 0.5 + k = 1.658312395 Solving 0.5 + k = 1.658312395 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 = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.658312395 + -0.5 k = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 k = 1.158312395 Simplifying k = 1.158312395Subproblem 2
k + 0.5 = -1.658312395 Simplifying k + 0.5 = -1.658312395 Reorder the terms: 0.5 + k = -1.658312395 Solving 0.5 + k = -1.658312395 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 = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.658312395 + -0.5 k = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 k = -2.158312395 Simplifying k = -2.158312395Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.158312395, -2.158312395}
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