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Simplifying 9 = 12k(3k + -4) + -7k Reorder the terms: 9 = 12k(-4 + 3k) + -7k 9 = (-4 * 12k + 3k * 12k) + -7k 9 = (-48k + 36k2) + -7k Reorder the terms: 9 = -48k + -7k + 36k2 Combine like terms: -48k + -7k = -55k 9 = -55k + 36k2 Solving 9 = -55k + 36k2 Solving for variable 'k'. Reorder the terms: 9 + 55k + -36k2 = -55k + 55k + 36k2 + -36k2 Combine like terms: -55k + 55k = 0 9 + 55k + -36k2 = 0 + 36k2 + -36k2 9 + 55k + -36k2 = 36k2 + -36k2 Combine like terms: 36k2 + -36k2 = 0 9 + 55k + -36k2 = 0 Begin completing the square. Divide all terms by -36 the coefficient of the squared term: Divide each side by '-36'. -0.25 + -1.527777778k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.527777778k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.527777778k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.527777778k + k2 = 0 + 0.25 -1.527777778k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.527777778k + k2 = 0.25 The k term is -1.527777778k. Take half its coefficient (-0.763888889). Square it (0.5835262347) and add it to both sides. Add '0.5835262347' to each side of the equation. -1.527777778k + 0.5835262347 + k2 = 0.25 + 0.5835262347 Reorder the terms: 0.5835262347 + -1.527777778k + k2 = 0.25 + 0.5835262347 Combine like terms: 0.25 + 0.5835262347 = 0.8335262347 0.5835262347 + -1.527777778k + k2 = 0.8335262347 Factor a perfect square on the left side: (k + -0.763888889)(k + -0.763888889) = 0.8335262347 Calculate the square root of the right side: 0.912976579 Break this problem into two subproblems by setting (k + -0.763888889) equal to 0.912976579 and -0.912976579.Subproblem 1
k + -0.763888889 = 0.912976579 Simplifying k + -0.763888889 = 0.912976579 Reorder the terms: -0.763888889 + k = 0.912976579 Solving -0.763888889 + k = 0.912976579 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.763888889' to each side of the equation. -0.763888889 + 0.763888889 + k = 0.912976579 + 0.763888889 Combine like terms: -0.763888889 + 0.763888889 = 0.000000000 0.000000000 + k = 0.912976579 + 0.763888889 k = 0.912976579 + 0.763888889 Combine like terms: 0.912976579 + 0.763888889 = 1.676865468 k = 1.676865468 Simplifying k = 1.676865468Subproblem 2
k + -0.763888889 = -0.912976579 Simplifying k + -0.763888889 = -0.912976579 Reorder the terms: -0.763888889 + k = -0.912976579 Solving -0.763888889 + k = -0.912976579 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.763888889' to each side of the equation. -0.763888889 + 0.763888889 + k = -0.912976579 + 0.763888889 Combine like terms: -0.763888889 + 0.763888889 = 0.000000000 0.000000000 + k = -0.912976579 + 0.763888889 k = -0.912976579 + 0.763888889 Combine like terms: -0.912976579 + 0.763888889 = -0.14908769 k = -0.14908769 Simplifying k = -0.14908769Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.676865468, -0.14908769}
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