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Simplifying (k + -3)(k + -2) + -4 = 0 Reorder the terms: (-3 + k)(k + -2) + -4 = 0 Reorder the terms: (-3 + k)(-2 + k) + -4 = 0 Multiply (-3 + k) * (-2 + k) (-3(-2 + k) + k(-2 + k)) + -4 = 0 ((-2 * -3 + k * -3) + k(-2 + k)) + -4 = 0 ((6 + -3k) + k(-2 + k)) + -4 = 0 (6 + -3k + (-2 * k + k * k)) + -4 = 0 (6 + -3k + (-2k + k2)) + -4 = 0 Combine like terms: -3k + -2k = -5k (6 + -5k + k2) + -4 = 0 Reorder the terms: 6 + -4 + -5k + k2 = 0 Combine like terms: 6 + -4 = 2 2 + -5k + k2 = 0 Solving 2 + -5k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -5k + -2 + k2 = 0 + -2 Reorder the terms: 2 + -2 + -5k + k2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -5k + k2 = 0 + -2 -5k + k2 = 0 + -2 Combine like terms: 0 + -2 = -2 -5k + k2 = -2 The k term is -5k. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5k + 6.25 + k2 = -2 + 6.25 Reorder the terms: 6.25 + -5k + k2 = -2 + 6.25 Combine like terms: -2 + 6.25 = 4.25 6.25 + -5k + k2 = 4.25 Factor a perfect square on the left side: (k + -2.5)(k + -2.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (k + -2.5) equal to 2.061552813 and -2.061552813.Subproblem 1
k + -2.5 = 2.061552813 Simplifying k + -2.5 = 2.061552813 Reorder the terms: -2.5 + k = 2.061552813 Solving -2.5 + k = 2.061552813 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + k = 2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = 2.061552813 + 2.5 k = 2.061552813 + 2.5 Combine like terms: 2.061552813 + 2.5 = 4.561552813 k = 4.561552813 Simplifying k = 4.561552813Subproblem 2
k + -2.5 = -2.061552813 Simplifying k + -2.5 = -2.061552813 Reorder the terms: -2.5 + k = -2.061552813 Solving -2.5 + k = -2.061552813 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + k = -2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = -2.061552813 + 2.5 k = -2.061552813 + 2.5 Combine like terms: -2.061552813 + 2.5 = 0.438447187 k = 0.438447187 Simplifying k = 0.438447187Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.561552813, 0.438447187}
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