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Simplifying 0 = 4k2 + 4k + -27 Reorder the terms: 0 = -27 + 4k + 4k2 Solving 0 = -27 + 4k + 4k2 Solving for variable 'k'. Combine like terms: 0 + 27 = 27 27 + -4k + -4k2 = -27 + 4k + 4k2 + 27 + -4k + -4k2 Reorder the terms: 27 + -4k + -4k2 = -27 + 27 + 4k + -4k + 4k2 + -4k2 Combine like terms: -27 + 27 = 0 27 + -4k + -4k2 = 0 + 4k + -4k + 4k2 + -4k2 27 + -4k + -4k2 = 4k + -4k + 4k2 + -4k2 Combine like terms: 4k + -4k = 0 27 + -4k + -4k2 = 0 + 4k2 + -4k2 27 + -4k + -4k2 = 4k2 + -4k2 Combine like terms: 4k2 + -4k2 = 0 27 + -4k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -6.75 + k + k2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + k + 6.75 + k2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + k + k2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + k + k2 = 0 + 6.75 k + k2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 k + k2 = 6.75 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. + 0.25 + k2 = 6.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + k2 = 6.75 + 0.25 Combine like terms: 6.75 + 0.25 = 7 1.25 + k2 = 7 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (k + 0.5) equal to 2.645751311 and -2.645751311.Subproblem 1
k + 0.5 = 2.645751311 Simplifying k + 0.5 = 2.645751311 Reorder the terms: 0.5 + k = 2.645751311 Solving 0.5 + k = 2.645751311 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 = 2.645751311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 2.645751311 + -0.5 k = 2.645751311 + -0.5 Combine like terms: 2.645751311 + -0.5 = 2.145751311 k = 2.145751311 Simplifying k = 2.145751311Subproblem 2
k + 0.5 = -2.645751311 Simplifying k + 0.5 = -2.645751311 Reorder the terms: 0.5 + k = -2.645751311 Solving 0.5 + k = -2.645751311 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 = -2.645751311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -2.645751311 + -0.5 k = -2.645751311 + -0.5 Combine like terms: -2.645751311 + -0.5 = -3.145751311 k = -3.145751311 Simplifying k = -3.145751311Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.145751311, -3.145751311}
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