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Simplifying 5k2 + -5k + -14 = -12 Reorder the terms: -14 + -5k + 5k2 = -12 Solving -14 + -5k + 5k2 = -12 Solving for variable 'k'. Reorder the terms: -14 + 12 + -5k + 5k2 = -12 + 12 Combine like terms: -14 + 12 = -2 -2 + -5k + 5k2 = -12 + 12 Combine like terms: -12 + 12 = 0 -2 + -5k + 5k2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + -1k + k2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -1k + 0.4 + k2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -1k + k2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -1k + k2 = 0 + 0.4 -1k + k2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -1k + k2 = 0.4 The k term is -1k. 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. -1k + 0.25 + k2 = 0.4 + 0.25 Reorder the terms: 0.25 + -1k + k2 = 0.4 + 0.25 Combine like terms: 0.4 + 0.25 = 0.65 0.25 + -1k + k2 = 0.65 Factor a perfect square on the left side: (k + -0.5)(k + -0.5) = 0.65 Calculate the square root of the right side: 0.806225775 Break this problem into two subproblems by setting (k + -0.5) equal to 0.806225775 and -0.806225775.Subproblem 1
k + -0.5 = 0.806225775 Simplifying k + -0.5 = 0.806225775 Reorder the terms: -0.5 + k = 0.806225775 Solving -0.5 + k = 0.806225775 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 = 0.806225775 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = 0.806225775 + 0.5 k = 0.806225775 + 0.5 Combine like terms: 0.806225775 + 0.5 = 1.306225775 k = 1.306225775 Simplifying k = 1.306225775Subproblem 2
k + -0.5 = -0.806225775 Simplifying k + -0.5 = -0.806225775 Reorder the terms: -0.5 + k = -0.806225775 Solving -0.5 + k = -0.806225775 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 = -0.806225775 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = -0.806225775 + 0.5 k = -0.806225775 + 0.5 Combine like terms: -0.806225775 + 0.5 = -0.306225775 k = -0.306225775 Simplifying k = -0.306225775Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.306225775, -0.306225775}
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