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Simplifying (3k)(3k) = 8k + 8 Remove parenthesis around (3k) 3k(3k) = 8k + 8 Remove parenthesis around (3k) 3k * 3k = 8k + 8 Reorder the terms for easier multiplication: 3 * 3k * k = 8k + 8 Multiply 3 * 3 9k * k = 8k + 8 Multiply k * k 9k2 = 8k + 8 Reorder the terms: 9k2 = 8 + 8k Solving 9k2 = 8 + 8k Solving for variable 'k'. Reorder the terms: -8 + -8k + 9k2 = 8 + 8k + -8 + -8k Reorder the terms: -8 + -8k + 9k2 = 8 + -8 + 8k + -8k Combine like terms: 8 + -8 = 0 -8 + -8k + 9k2 = 0 + 8k + -8k -8 + -8k + 9k2 = 8k + -8k Combine like terms: 8k + -8k = 0 -8 + -8k + 9k2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.8888888889 + -0.8888888889k + k2 = 0 Move the constant term to the right: Add '0.8888888889' to each side of the equation. -0.8888888889 + -0.8888888889k + 0.8888888889 + k2 = 0 + 0.8888888889 Reorder the terms: -0.8888888889 + 0.8888888889 + -0.8888888889k + k2 = 0 + 0.8888888889 Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000 0.0000000000 + -0.8888888889k + k2 = 0 + 0.8888888889 -0.8888888889k + k2 = 0 + 0.8888888889 Combine like terms: 0 + 0.8888888889 = 0.8888888889 -0.8888888889k + k2 = 0.8888888889 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 = 0.8888888889 + 0.1975308642 Reorder the terms: 0.1975308642 + -0.8888888889k + k2 = 0.8888888889 + 0.1975308642 Combine like terms: 0.8888888889 + 0.1975308642 = 1.0864197531 0.1975308642 + -0.8888888889k + k2 = 1.0864197531 Factor a perfect square on the left side: (k + -0.4444444445)(k + -0.4444444445) = 1.0864197531 Calculate the square root of the right side: 1.042314613 Break this problem into two subproblems by setting (k + -0.4444444445) equal to 1.042314613 and -1.042314613.Subproblem 1
k + -0.4444444445 = 1.042314613 Simplifying k + -0.4444444445 = 1.042314613 Reorder the terms: -0.4444444445 + k = 1.042314613 Solving -0.4444444445 + k = 1.042314613 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 = 1.042314613 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + k = 1.042314613 + 0.4444444445 k = 1.042314613 + 0.4444444445 Combine like terms: 1.042314613 + 0.4444444445 = 1.4867590575 k = 1.4867590575 Simplifying k = 1.4867590575Subproblem 2
k + -0.4444444445 = -1.042314613 Simplifying k + -0.4444444445 = -1.042314613 Reorder the terms: -0.4444444445 + k = -1.042314613 Solving -0.4444444445 + k = -1.042314613 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 = -1.042314613 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + k = -1.042314613 + 0.4444444445 k = -1.042314613 + 0.4444444445 Combine like terms: -1.042314613 + 0.4444444445 = -0.5978701685 k = -0.5978701685 Simplifying k = -0.5978701685Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.4867590575, -0.5978701685}
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