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Simplifying 9k2 + -4k + -16 = 0 Reorder the terms: -16 + -4k + 9k2 = 0 Solving -16 + -4k + 9k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + -0.4444444444k + k2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + -0.4444444444k + 1.777777778 + k2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + -0.4444444444k + k2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + -0.4444444444k + k2 = 0 + 1.777777778 -0.4444444444k + k2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 -0.4444444444k + k2 = 1.777777778 The k term is -0.4444444444k. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444k + 0.04938271604 + k2 = 1.777777778 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444k + k2 = 1.777777778 + 0.04938271604 Combine like terms: 1.777777778 + 0.04938271604 = 1.82716049404 0.04938271604 + -0.4444444444k + k2 = 1.82716049404 Factor a perfect square on the left side: (k + -0.2222222222)(k + -0.2222222222) = 1.82716049404 Calculate the square root of the right side: 1.351725007 Break this problem into two subproblems by setting (k + -0.2222222222) equal to 1.351725007 and -1.351725007.Subproblem 1
k + -0.2222222222 = 1.351725007 Simplifying k + -0.2222222222 = 1.351725007 Reorder the terms: -0.2222222222 + k = 1.351725007 Solving -0.2222222222 + k = 1.351725007 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + k = 1.351725007 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + k = 1.351725007 + 0.2222222222 k = 1.351725007 + 0.2222222222 Combine like terms: 1.351725007 + 0.2222222222 = 1.5739472292 k = 1.5739472292 Simplifying k = 1.5739472292Subproblem 2
k + -0.2222222222 = -1.351725007 Simplifying k + -0.2222222222 = -1.351725007 Reorder the terms: -0.2222222222 + k = -1.351725007 Solving -0.2222222222 + k = -1.351725007 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + k = -1.351725007 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + k = -1.351725007 + 0.2222222222 k = -1.351725007 + 0.2222222222 Combine like terms: -1.351725007 + 0.2222222222 = -1.1295027848 k = -1.1295027848 Simplifying k = -1.1295027848Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.5739472292, -1.1295027848}
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