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Simplifying 9 = 4(3k + -4) * 7k Reorder the terms: 9 = 4(-4 + 3k) * 7k Reorder the terms for easier multiplication: 9 = 4 * 7k(-4 + 3k) Multiply 4 * 7 9 = 28k(-4 + 3k) 9 = (-4 * 28k + 3k * 28k) 9 = (-112k + 84k2) Solving 9 = -112k + 84k2 Solving for variable 'k'. Reorder the terms: 9 + 112k + -84k2 = -112k + 112k + 84k2 + -84k2 Combine like terms: -112k + 112k = 0 9 + 112k + -84k2 = 0 + 84k2 + -84k2 9 + 112k + -84k2 = 84k2 + -84k2 Combine like terms: 84k2 + -84k2 = 0 9 + 112k + -84k2 = 0 Begin completing the square. Divide all terms by -84 the coefficient of the squared term: Divide each side by '-84'. -0.1071428571 + -1.333333333k + k2 = 0 Move the constant term to the right: Add '0.1071428571' to each side of the equation. -0.1071428571 + -1.333333333k + 0.1071428571 + k2 = 0 + 0.1071428571 Reorder the terms: -0.1071428571 + 0.1071428571 + -1.333333333k + k2 = 0 + 0.1071428571 Combine like terms: -0.1071428571 + 0.1071428571 = 0.0000000000 0.0000000000 + -1.333333333k + k2 = 0 + 0.1071428571 -1.333333333k + k2 = 0 + 0.1071428571 Combine like terms: 0 + 0.1071428571 = 0.1071428571 -1.333333333k + k2 = 0.1071428571 The k term is -1.333333333k. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333k + 0.4444444442 + k2 = 0.1071428571 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333k + k2 = 0.1071428571 + 0.4444444442 Combine like terms: 0.1071428571 + 0.4444444442 = 0.5515873013 0.4444444442 + -1.333333333k + k2 = 0.5515873013 Factor a perfect square on the left side: (k + -0.6666666665)(k + -0.6666666665) = 0.5515873013 Calculate the square root of the right side: 0.742689236 Break this problem into two subproblems by setting (k + -0.6666666665) equal to 0.742689236 and -0.742689236.Subproblem 1
k + -0.6666666665 = 0.742689236 Simplifying k + -0.6666666665 = 0.742689236 Reorder the terms: -0.6666666665 + k = 0.742689236 Solving -0.6666666665 + k = 0.742689236 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + k = 0.742689236 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + k = 0.742689236 + 0.6666666665 k = 0.742689236 + 0.6666666665 Combine like terms: 0.742689236 + 0.6666666665 = 1.4093559025 k = 1.4093559025 Simplifying k = 1.4093559025Subproblem 2
k + -0.6666666665 = -0.742689236 Simplifying k + -0.6666666665 = -0.742689236 Reorder the terms: -0.6666666665 + k = -0.742689236 Solving -0.6666666665 + k = -0.742689236 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + k = -0.742689236 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + k = -0.742689236 + 0.6666666665 k = -0.742689236 + 0.6666666665 Combine like terms: -0.742689236 + 0.6666666665 = -0.0760225695 k = -0.0760225695 Simplifying k = -0.0760225695Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.4093559025, -0.0760225695}
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