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Simplifying -8k(k + -4) + -1(-5k + -5) = -4k Reorder the terms: -8k(-4 + k) + -1(-5k + -5) = -4k (-4 * -8k + k * -8k) + -1(-5k + -5) = -4k (32k + -8k2) + -1(-5k + -5) = -4k Reorder the terms: 32k + -8k2 + -1(-5 + -5k) = -4k 32k + -8k2 + (-5 * -1 + -5k * -1) = -4k 32k + -8k2 + (5 + 5k) = -4k Reorder the terms: 5 + 32k + 5k + -8k2 = -4k Combine like terms: 32k + 5k = 37k 5 + 37k + -8k2 = -4k Solving 5 + 37k + -8k2 = -4k Solving for variable 'k'. Reorder the terms: 5 + 37k + 4k + -8k2 = -4k + 4k Combine like terms: 37k + 4k = 41k 5 + 41k + -8k2 = -4k + 4k Combine like terms: -4k + 4k = 0 5 + 41k + -8k2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.625 + -5.125k + k2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -5.125k + 0.625 + k2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -5.125k + k2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -5.125k + k2 = 0 + 0.625 -5.125k + k2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -5.125k + k2 = 0.625 The k term is -5.125k. Take half its coefficient (-2.5625). Square it (6.56640625) and add it to both sides. Add '6.56640625' to each side of the equation. -5.125k + 6.56640625 + k2 = 0.625 + 6.56640625 Reorder the terms: 6.56640625 + -5.125k + k2 = 0.625 + 6.56640625 Combine like terms: 0.625 + 6.56640625 = 7.19140625 6.56640625 + -5.125k + k2 = 7.19140625 Factor a perfect square on the left side: (k + -2.5625)(k + -2.5625) = 7.19140625 Calculate the square root of the right side: 2.681679744 Break this problem into two subproblems by setting (k + -2.5625) equal to 2.681679744 and -2.681679744.Subproblem 1
k + -2.5625 = 2.681679744 Simplifying k + -2.5625 = 2.681679744 Reorder the terms: -2.5625 + k = 2.681679744 Solving -2.5625 + k = 2.681679744 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5625' to each side of the equation. -2.5625 + 2.5625 + k = 2.681679744 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + k = 2.681679744 + 2.5625 k = 2.681679744 + 2.5625 Combine like terms: 2.681679744 + 2.5625 = 5.244179744 k = 5.244179744 Simplifying k = 5.244179744Subproblem 2
k + -2.5625 = -2.681679744 Simplifying k + -2.5625 = -2.681679744 Reorder the terms: -2.5625 + k = -2.681679744 Solving -2.5625 + k = -2.681679744 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5625' to each side of the equation. -2.5625 + 2.5625 + k = -2.681679744 + 2.5625 Combine like terms: -2.5625 + 2.5625 = 0.0000 0.0000 + k = -2.681679744 + 2.5625 k = -2.681679744 + 2.5625 Combine like terms: -2.681679744 + 2.5625 = -0.119179744 k = -0.119179744 Simplifying k = -0.119179744Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.244179744, -0.119179744}
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