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Simplifying 9k2 + 2k + -16 = 0 Reorder the terms: -16 + 2k + 9k2 = 0 Solving -16 + 2k + 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 + 0.2222222222k + k2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 0.2222222222k + 1.777777778 + k2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 0.2222222222k + k2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 0.2222222222k + k2 = 0 + 1.777777778 0.2222222222k + k2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 0.2222222222k + k2 = 1.777777778 The k term is 0.2222222222k. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222k + 0.01234567901 + k2 = 1.777777778 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222k + k2 = 1.777777778 + 0.01234567901 Combine like terms: 1.777777778 + 0.01234567901 = 1.79012345701 0.01234567901 + 0.2222222222k + k2 = 1.79012345701 Factor a perfect square on the left side: (k + 0.1111111111)(k + 0.1111111111) = 1.79012345701 Calculate the square root of the right side: 1.337954953 Break this problem into two subproblems by setting (k + 0.1111111111) equal to 1.337954953 and -1.337954953.Subproblem 1
k + 0.1111111111 = 1.337954953 Simplifying k + 0.1111111111 = 1.337954953 Reorder the terms: 0.1111111111 + k = 1.337954953 Solving 0.1111111111 + k = 1.337954953 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + k = 1.337954953 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + k = 1.337954953 + -0.1111111111 k = 1.337954953 + -0.1111111111 Combine like terms: 1.337954953 + -0.1111111111 = 1.2268438419 k = 1.2268438419 Simplifying k = 1.2268438419Subproblem 2
k + 0.1111111111 = -1.337954953 Simplifying k + 0.1111111111 = -1.337954953 Reorder the terms: 0.1111111111 + k = -1.337954953 Solving 0.1111111111 + k = -1.337954953 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + k = -1.337954953 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + k = -1.337954953 + -0.1111111111 k = -1.337954953 + -0.1111111111 Combine like terms: -1.337954953 + -0.1111111111 = -1.4490660641 k = -1.4490660641 Simplifying k = -1.4490660641Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.2268438419, -1.4490660641}
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