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Simplifying k(2k + 3) = 19 Reorder the terms: k(3 + 2k) = 19 (3 * k + 2k * k) = 19 (3k + 2k2) = 19 Solving 3k + 2k2 = 19 Solving for variable 'k'. Reorder the terms: -19 + 3k + 2k2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 3k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + 1.5k + k2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 1.5k + 9.5 + k2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 1.5k + k2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 1.5k + k2 = 0 + 9.5 1.5k + k2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 1.5k + k2 = 9.5 The k term is 1.5k. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5k + 0.5625 + k2 = 9.5 + 0.5625 Reorder the terms: 0.5625 + 1.5k + k2 = 9.5 + 0.5625 Combine like terms: 9.5 + 0.5625 = 10.0625 0.5625 + 1.5k + k2 = 10.0625 Factor a perfect square on the left side: (k + 0.75)(k + 0.75) = 10.0625 Calculate the square root of the right side: 3.172144385 Break this problem into two subproblems by setting (k + 0.75) equal to 3.172144385 and -3.172144385.Subproblem 1
k + 0.75 = 3.172144385 Simplifying k + 0.75 = 3.172144385 Reorder the terms: 0.75 + k = 3.172144385 Solving 0.75 + k = 3.172144385 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + k = 3.172144385 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = 3.172144385 + -0.75 k = 3.172144385 + -0.75 Combine like terms: 3.172144385 + -0.75 = 2.422144385 k = 2.422144385 Simplifying k = 2.422144385Subproblem 2
k + 0.75 = -3.172144385 Simplifying k + 0.75 = -3.172144385 Reorder the terms: 0.75 + k = -3.172144385 Solving 0.75 + k = -3.172144385 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + k = -3.172144385 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + k = -3.172144385 + -0.75 k = -3.172144385 + -0.75 Combine like terms: -3.172144385 + -0.75 = -3.922144385 k = -3.922144385 Simplifying k = -3.922144385Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.422144385, -3.922144385}
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