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Simplifying 4k2 + -12 + 7k + -2k2 + -16k = 0 Reorder the terms: -12 + 7k + -16k + 4k2 + -2k2 = 0 Combine like terms: 7k + -16k = -9k -12 + -9k + 4k2 + -2k2 = 0 Combine like terms: 4k2 + -2k2 = 2k2 -12 + -9k + 2k2 = 0 Solving -12 + -9k + 2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + -4.5k + k2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -4.5k + 6 + k2 = 0 + 6 Reorder the terms: -6 + 6 + -4.5k + k2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -4.5k + k2 = 0 + 6 -4.5k + k2 = 0 + 6 Combine like terms: 0 + 6 = 6 -4.5k + k2 = 6 The k term is -4.5k. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5k + 5.0625 + k2 = 6 + 5.0625 Reorder the terms: 5.0625 + -4.5k + k2 = 6 + 5.0625 Combine like terms: 6 + 5.0625 = 11.0625 5.0625 + -4.5k + k2 = 11.0625 Factor a perfect square on the left side: (k + -2.25)(k + -2.25) = 11.0625 Calculate the square root of the right side: 3.326033674 Break this problem into two subproblems by setting (k + -2.25) equal to 3.326033674 and -3.326033674.Subproblem 1
k + -2.25 = 3.326033674 Simplifying k + -2.25 = 3.326033674 Reorder the terms: -2.25 + k = 3.326033674 Solving -2.25 + k = 3.326033674 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + k = 3.326033674 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + k = 3.326033674 + 2.25 k = 3.326033674 + 2.25 Combine like terms: 3.326033674 + 2.25 = 5.576033674 k = 5.576033674 Simplifying k = 5.576033674Subproblem 2
k + -2.25 = -3.326033674 Simplifying k + -2.25 = -3.326033674 Reorder the terms: -2.25 + k = -3.326033674 Solving -2.25 + k = -3.326033674 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + k = -3.326033674 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + k = -3.326033674 + 2.25 k = -3.326033674 + 2.25 Combine like terms: -3.326033674 + 2.25 = -1.076033674 k = -1.076033674 Simplifying k = -1.076033674Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.576033674, -1.076033674}
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