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Simplifying 9k2 + 9k + 1 = 0 Reorder the terms: 1 + 9k + 9k2 = 0 Solving 1 + 9k + 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'. 0.1111111111 + k + k2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + k + -0.1111111111 + k2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + k + k2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + k + k2 = 0 + -0.1111111111 k + k2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 k + k2 = -0.1111111111 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + k2 = -0.1111111111 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + k2 = -0.1111111111 + 0.25 Combine like terms: -0.1111111111 + 0.25 = 0.1388888889 1.25 + k2 = 0.1388888889 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 0.1388888889 Calculate the square root of the right side: 0.372677996 Break this problem into two subproblems by setting (k + 0.5) equal to 0.372677996 and -0.372677996.Subproblem 1
k + 0.5 = 0.372677996 Simplifying k + 0.5 = 0.372677996 Reorder the terms: 0.5 + k = 0.372677996 Solving 0.5 + k = 0.372677996 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 0.372677996 + -0.5 k = 0.372677996 + -0.5 Combine like terms: 0.372677996 + -0.5 = -0.127322004 k = -0.127322004 Simplifying k = -0.127322004Subproblem 2
k + 0.5 = -0.372677996 Simplifying k + 0.5 = -0.372677996 Reorder the terms: 0.5 + k = -0.372677996 Solving 0.5 + k = -0.372677996 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -0.372677996 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -0.372677996 + -0.5 k = -0.372677996 + -0.5 Combine like terms: -0.372677996 + -0.5 = -0.872677996 k = -0.872677996 Simplifying k = -0.872677996Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.127322004, -0.872677996}
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