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Simplifying 9k2 + 2k + -5 = 0 Reorder the terms: -5 + 2k + 9k2 = 0 Solving -5 + 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'. -0.5555555556 + 0.2222222222k + k2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + 0.2222222222k + 0.5555555556 + k2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + 0.2222222222k + k2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + 0.2222222222k + k2 = 0 + 0.5555555556 0.2222222222k + k2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 0.2222222222k + k2 = 0.5555555556 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 = 0.5555555556 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222k + k2 = 0.5555555556 + 0.01234567901 Combine like terms: 0.5555555556 + 0.01234567901 = 0.56790123461 0.01234567901 + 0.2222222222k + k2 = 0.56790123461 Factor a perfect square on the left side: (k + 0.1111111111)(k + 0.1111111111) = 0.56790123461 Calculate the square root of the right side: 0.75359222 Break this problem into two subproblems by setting (k + 0.1111111111) equal to 0.75359222 and -0.75359222.Subproblem 1
k + 0.1111111111 = 0.75359222 Simplifying k + 0.1111111111 = 0.75359222 Reorder the terms: 0.1111111111 + k = 0.75359222 Solving 0.1111111111 + k = 0.75359222 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 = 0.75359222 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + k = 0.75359222 + -0.1111111111 k = 0.75359222 + -0.1111111111 Combine like terms: 0.75359222 + -0.1111111111 = 0.6424811089 k = 0.6424811089 Simplifying k = 0.6424811089Subproblem 2
k + 0.1111111111 = -0.75359222 Simplifying k + 0.1111111111 = -0.75359222 Reorder the terms: 0.1111111111 + k = -0.75359222 Solving 0.1111111111 + k = -0.75359222 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 = -0.75359222 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + k = -0.75359222 + -0.1111111111 k = -0.75359222 + -0.1111111111 Combine like terms: -0.75359222 + -0.1111111111 = -0.8647033311 k = -0.8647033311 Simplifying k = -0.8647033311Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.6424811089, -0.8647033311}
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