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Simplifying -4k(k + 6) + -1(9k + -6) = -14k Reorder the terms: -4k(6 + k) + -1(9k + -6) = -14k (6 * -4k + k * -4k) + -1(9k + -6) = -14k (-24k + -4k2) + -1(9k + -6) = -14k Reorder the terms: -24k + -4k2 + -1(-6 + 9k) = -14k -24k + -4k2 + (-6 * -1 + 9k * -1) = -14k -24k + -4k2 + (6 + -9k) = -14k Reorder the terms: 6 + -24k + -9k + -4k2 = -14k Combine like terms: -24k + -9k = -33k 6 + -33k + -4k2 = -14k Solving 6 + -33k + -4k2 = -14k Solving for variable 'k'. Reorder the terms: 6 + -33k + 14k + -4k2 = -14k + 14k Combine like terms: -33k + 14k = -19k 6 + -19k + -4k2 = -14k + 14k Combine like terms: -14k + 14k = 0 6 + -19k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.5 + 4.75k + k2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 4.75k + 1.5 + k2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 4.75k + k2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 4.75k + k2 = 0 + 1.5 4.75k + k2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 4.75k + k2 = 1.5 The k term is 4.75k. Take half its coefficient (2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. 4.75k + 5.640625 + k2 = 1.5 + 5.640625 Reorder the terms: 5.640625 + 4.75k + k2 = 1.5 + 5.640625 Combine like terms: 1.5 + 5.640625 = 7.140625 5.640625 + 4.75k + k2 = 7.140625 Factor a perfect square on the left side: (k + 2.375)(k + 2.375) = 7.140625 Calculate the square root of the right side: 2.672194791 Break this problem into two subproblems by setting (k + 2.375) equal to 2.672194791 and -2.672194791.Subproblem 1
k + 2.375 = 2.672194791 Simplifying k + 2.375 = 2.672194791 Reorder the terms: 2.375 + k = 2.672194791 Solving 2.375 + k = 2.672194791 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + k = 2.672194791 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + k = 2.672194791 + -2.375 k = 2.672194791 + -2.375 Combine like terms: 2.672194791 + -2.375 = 0.297194791 k = 0.297194791 Simplifying k = 0.297194791Subproblem 2
k + 2.375 = -2.672194791 Simplifying k + 2.375 = -2.672194791 Reorder the terms: 2.375 + k = -2.672194791 Solving 2.375 + k = -2.672194791 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + k = -2.672194791 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + k = -2.672194791 + -2.375 k = -2.672194791 + -2.375 Combine like terms: -2.672194791 + -2.375 = -5.047194791 k = -5.047194791 Simplifying k = -5.047194791Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.297194791, -5.047194791}
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