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Simplifying 4k(3 + k) = -1(2k + 3) (3 * 4k + k * 4k) = -1(2k + 3) (12k + 4k2) = -1(2k + 3) Reorder the terms: 12k + 4k2 = -1(3 + 2k) 12k + 4k2 = (3 * -1 + 2k * -1) 12k + 4k2 = (-3 + -2k) Solving 12k + 4k2 = -3 + -2k Solving for variable 'k'. Reorder the terms: 3 + 12k + 2k + 4k2 = -3 + -2k + 3 + 2k Combine like terms: 12k + 2k = 14k 3 + 14k + 4k2 = -3 + -2k + 3 + 2k Reorder the terms: 3 + 14k + 4k2 = -3 + 3 + -2k + 2k Combine like terms: -3 + 3 = 0 3 + 14k + 4k2 = 0 + -2k + 2k 3 + 14k + 4k2 = -2k + 2k Combine like terms: -2k + 2k = 0 3 + 14k + 4k2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 3.5k + k2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 3.5k + -0.75 + k2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 3.5k + k2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 3.5k + k2 = 0 + -0.75 3.5k + k2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 3.5k + k2 = -0.75 The k term is 3.5k. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5k + 3.0625 + k2 = -0.75 + 3.0625 Reorder the terms: 3.0625 + 3.5k + k2 = -0.75 + 3.0625 Combine like terms: -0.75 + 3.0625 = 2.3125 3.0625 + 3.5k + k2 = 2.3125 Factor a perfect square on the left side: (k + 1.75)(k + 1.75) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (k + 1.75) equal to 1.520690633 and -1.520690633.Subproblem 1
k + 1.75 = 1.520690633 Simplifying k + 1.75 = 1.520690633 Reorder the terms: 1.75 + k = 1.520690633 Solving 1.75 + k = 1.520690633 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = 1.520690633 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = 1.520690633 + -1.75 k = 1.520690633 + -1.75 Combine like terms: 1.520690633 + -1.75 = -0.229309367 k = -0.229309367 Simplifying k = -0.229309367Subproblem 2
k + 1.75 = -1.520690633 Simplifying k + 1.75 = -1.520690633 Reorder the terms: 1.75 + k = -1.520690633 Solving 1.75 + k = -1.520690633 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = -1.520690633 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = -1.520690633 + -1.75 k = -1.520690633 + -1.75 Combine like terms: -1.520690633 + -1.75 = -3.270690633 k = -3.270690633 Simplifying k = -3.270690633Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.229309367, -3.270690633}
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