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Simplifying 12k2 + 4k = 21 + -7k Reorder the terms: 4k + 12k2 = 21 + -7k Solving 4k + 12k2 = 21 + -7k Solving for variable 'k'. Reorder the terms: -21 + 4k + 7k + 12k2 = 21 + -7k + -21 + 7k Combine like terms: 4k + 7k = 11k -21 + 11k + 12k2 = 21 + -7k + -21 + 7k Reorder the terms: -21 + 11k + 12k2 = 21 + -21 + -7k + 7k Combine like terms: 21 + -21 = 0 -21 + 11k + 12k2 = 0 + -7k + 7k -21 + 11k + 12k2 = -7k + 7k Combine like terms: -7k + 7k = 0 -21 + 11k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.75 + 0.9166666667k + k2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 0.9166666667k + 1.75 + k2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 0.9166666667k + k2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 0.9166666667k + k2 = 0 + 1.75 0.9166666667k + k2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 0.9166666667k + k2 = 1.75 The k term is 0.9166666667k. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667k + 0.2100694445 + k2 = 1.75 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667k + k2 = 1.75 + 0.2100694445 Combine like terms: 1.75 + 0.2100694445 = 1.9600694445 0.2100694445 + 0.9166666667k + k2 = 1.9600694445 Factor a perfect square on the left side: (k + 0.4583333334)(k + 0.4583333334) = 1.9600694445 Calculate the square root of the right side: 1.400024801 Break this problem into two subproblems by setting (k + 0.4583333334) equal to 1.400024801 and -1.400024801.Subproblem 1
k + 0.4583333334 = 1.400024801 Simplifying k + 0.4583333334 = 1.400024801 Reorder the terms: 0.4583333334 + k = 1.400024801 Solving 0.4583333334 + k = 1.400024801 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + k = 1.400024801 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + k = 1.400024801 + -0.4583333334 k = 1.400024801 + -0.4583333334 Combine like terms: 1.400024801 + -0.4583333334 = 0.9416914676 k = 0.9416914676 Simplifying k = 0.9416914676Subproblem 2
k + 0.4583333334 = -1.400024801 Simplifying k + 0.4583333334 = -1.400024801 Reorder the terms: 0.4583333334 + k = -1.400024801 Solving 0.4583333334 + k = -1.400024801 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + k = -1.400024801 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + k = -1.400024801 + -0.4583333334 k = -1.400024801 + -0.4583333334 Combine like terms: -1.400024801 + -0.4583333334 = -1.8583581344 k = -1.8583581344 Simplifying k = -1.8583581344Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.9416914676, -1.8583581344}
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