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Simplifying 3k(-6 + -18k) + 17k + -3 = 0 (-6 * 3k + -18k * 3k) + 17k + -3 = 0 (-18k + -54k2) + 17k + -3 = 0 Reorder the terms: -3 + -18k + 17k + -54k2 = 0 Combine like terms: -18k + 17k = -1k -3 + -1k + -54k2 = 0 Solving -3 + -1k + -54k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '-1'. -1(3 + k + 54k2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + k + 54k2)' equal to zero and attempt to solve: Simplifying 3 + k + 54k2 = 0 Solving 3 + k + 54k2 = 0 Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. 0.05555555556 + 0.01851851852k + k2 = 0 Move the constant term to the right: Add '-0.05555555556' to each side of the equation. 0.05555555556 + 0.01851851852k + -0.05555555556 + k2 = 0 + -0.05555555556 Reorder the terms: 0.05555555556 + -0.05555555556 + 0.01851851852k + k2 = 0 + -0.05555555556 Combine like terms: 0.05555555556 + -0.05555555556 = 0.00000000000 0.00000000000 + 0.01851851852k + k2 = 0 + -0.05555555556 0.01851851852k + k2 = 0 + -0.05555555556 Combine like terms: 0 + -0.05555555556 = -0.05555555556 0.01851851852k + k2 = -0.05555555556 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. 0.01851851852k + 0.25 + k2 = -0.05555555556 + 0.25 Reorder the terms: 0.25 + 0.01851851852k + k2 = -0.05555555556 + 0.25 Combine like terms: -0.05555555556 + 0.25 = 0.19444444444 0.25 + 0.01851851852k + k2 = 0.19444444444 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 0.19444444444 Calculate the square root of the right side: 0.440958552 Break this problem into two subproblems by setting (k + 0.5) equal to 0.440958552 and -0.440958552.Subproblem 1
k + 0.5 = 0.440958552 Simplifying k + 0.5 = 0.440958552 Reorder the terms: 0.5 + k = 0.440958552 Solving 0.5 + k = 0.440958552 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 = 0.440958552 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 0.440958552 + -0.5 k = 0.440958552 + -0.5 Combine like terms: 0.440958552 + -0.5 = -0.059041448 k = -0.059041448 Simplifying k = -0.059041448Subproblem 2
k + 0.5 = -0.440958552 Simplifying k + 0.5 = -0.440958552 Reorder the terms: 0.5 + k = -0.440958552 Solving 0.5 + k = -0.440958552 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 = -0.440958552 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -0.440958552 + -0.5 k = -0.440958552 + -0.5 Combine like terms: -0.440958552 + -0.5 = -0.940958552 k = -0.940958552 Simplifying k = -0.940958552Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.059041448, -0.940958552}Solution
k = {-0.059041448, -0.940958552}
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