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Simplifying 38 + 7k = 8k(k + 4) Reorder the terms: 38 + 7k = 8k(4 + k) 38 + 7k = (4 * 8k + k * 8k) 38 + 7k = (32k + 8k2) Solving 38 + 7k = 32k + 8k2 Solving for variable 'k'. Combine like terms: 7k + -32k = -25k 38 + -25k + -8k2 = 32k + 8k2 + -32k + -8k2 Reorder the terms: 38 + -25k + -8k2 = 32k + -32k + 8k2 + -8k2 Combine like terms: 32k + -32k = 0 38 + -25k + -8k2 = 0 + 8k2 + -8k2 38 + -25k + -8k2 = 8k2 + -8k2 Combine like terms: 8k2 + -8k2 = 0 38 + -25k + -8k2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -4.75 + 3.125k + k2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + 3.125k + 4.75 + k2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + 3.125k + k2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + 3.125k + k2 = 0 + 4.75 3.125k + k2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 3.125k + k2 = 4.75 The k term is 3.125k. Take half its coefficient (1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. 3.125k + 2.44140625 + k2 = 4.75 + 2.44140625 Reorder the terms: 2.44140625 + 3.125k + k2 = 4.75 + 2.44140625 Combine like terms: 4.75 + 2.44140625 = 7.19140625 2.44140625 + 3.125k + k2 = 7.19140625 Factor a perfect square on the left side: (k + 1.5625)(k + 1.5625) = 7.19140625 Calculate the square root of the right side: 2.681679744 Break this problem into two subproblems by setting (k + 1.5625) equal to 2.681679744 and -2.681679744.Subproblem 1
k + 1.5625 = 2.681679744 Simplifying k + 1.5625 = 2.681679744 Reorder the terms: 1.5625 + k = 2.681679744 Solving 1.5625 + k = 2.681679744 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + k = 2.681679744 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + k = 2.681679744 + -1.5625 k = 2.681679744 + -1.5625 Combine like terms: 2.681679744 + -1.5625 = 1.119179744 k = 1.119179744 Simplifying k = 1.119179744Subproblem 2
k + 1.5625 = -2.681679744 Simplifying k + 1.5625 = -2.681679744 Reorder the terms: 1.5625 + k = -2.681679744 Solving 1.5625 + k = -2.681679744 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + k = -2.681679744 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + k = -2.681679744 + -1.5625 k = -2.681679744 + -1.5625 Combine like terms: -2.681679744 + -1.5625 = -4.244179744 k = -4.244179744 Simplifying k = -4.244179744Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.119179744, -4.244179744}
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