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Simplifying 9k2 + -8k + -5 = 0 Reorder the terms: -5 + -8k + 9k2 = 0 Solving -5 + -8k + 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 + -0.8888888889k + k2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + -0.8888888889k + 0.5555555556 + k2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + -0.8888888889k + k2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + -0.8888888889k + k2 = 0 + 0.5555555556 -0.8888888889k + k2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 -0.8888888889k + k2 = 0.5555555556 The k term is -0.8888888889k. Take half its coefficient (-0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. -0.8888888889k + 0.1975308642 + k2 = 0.5555555556 + 0.1975308642 Reorder the terms: 0.1975308642 + -0.8888888889k + k2 = 0.5555555556 + 0.1975308642 Combine like terms: 0.5555555556 + 0.1975308642 = 0.7530864198 0.1975308642 + -0.8888888889k + k2 = 0.7530864198 Factor a perfect square on the left side: (k + -0.4444444445)(k + -0.4444444445) = 0.7530864198 Calculate the square root of the right side: 0.86780552 Break this problem into two subproblems by setting (k + -0.4444444445) equal to 0.86780552 and -0.86780552.Subproblem 1
k + -0.4444444445 = 0.86780552 Simplifying k + -0.4444444445 = 0.86780552 Reorder the terms: -0.4444444445 + k = 0.86780552 Solving -0.4444444445 + k = 0.86780552 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + k = 0.86780552 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + k = 0.86780552 + 0.4444444445 k = 0.86780552 + 0.4444444445 Combine like terms: 0.86780552 + 0.4444444445 = 1.3122499645 k = 1.3122499645 Simplifying k = 1.3122499645Subproblem 2
k + -0.4444444445 = -0.86780552 Simplifying k + -0.4444444445 = -0.86780552 Reorder the terms: -0.4444444445 + k = -0.86780552 Solving -0.4444444445 + k = -0.86780552 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + k = -0.86780552 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + k = -0.86780552 + 0.4444444445 k = -0.86780552 + 0.4444444445 Combine like terms: -0.86780552 + 0.4444444445 = -0.4233610755 k = -0.4233610755 Simplifying k = -0.4233610755Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.3122499645, -0.4233610755}
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