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Simplifying 2K(K + 3) = 9 Reorder the terms: 2K(3 + K) = 9 (3 * 2K + K * 2K) = 9 (6K + 2K2) = 9 Solving 6K + 2K2 = 9 Solving for variable 'K'. Reorder the terms: -9 + 6K + 2K2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 6K + 2K2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 3K + K2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 3K + 4.5 + K2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 3K + K2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 3K + K2 = 0 + 4.5 3K + K2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 3K + K2 = 4.5 The K term is 3K. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3K + 2.25 + K2 = 4.5 + 2.25 Reorder the terms: 2.25 + 3K + K2 = 4.5 + 2.25 Combine like terms: 4.5 + 2.25 = 6.75 2.25 + 3K + K2 = 6.75 Factor a perfect square on the left side: (K + 1.5)(K + 1.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (K + 1.5) equal to 2.598076211 and -2.598076211.Subproblem 1
K + 1.5 = 2.598076211 Simplifying K + 1.5 = 2.598076211 Reorder the terms: 1.5 + K = 2.598076211 Solving 1.5 + K = 2.598076211 Solving for variable 'K'. Move all terms containing K to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + K = 2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + K = 2.598076211 + -1.5 K = 2.598076211 + -1.5 Combine like terms: 2.598076211 + -1.5 = 1.098076211 K = 1.098076211 Simplifying K = 1.098076211Subproblem 2
K + 1.5 = -2.598076211 Simplifying K + 1.5 = -2.598076211 Reorder the terms: 1.5 + K = -2.598076211 Solving 1.5 + K = -2.598076211 Solving for variable 'K'. Move all terms containing K to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + K = -2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + K = -2.598076211 + -1.5 K = -2.598076211 + -1.5 Combine like terms: -2.598076211 + -1.5 = -4.098076211 K = -4.098076211 Simplifying K = -4.098076211Solution
The solution to the problem is based on the solutions from the subproblems. K = {1.098076211, -4.098076211}
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