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Simplifying -8k(k + 6) = -5k + 2(k + -4) Reorder the terms: -8k(6 + k) = -5k + 2(k + -4) (6 * -8k + k * -8k) = -5k + 2(k + -4) (-48k + -8k2) = -5k + 2(k + -4) Reorder the terms: -48k + -8k2 = -5k + 2(-4 + k) -48k + -8k2 = -5k + (-4 * 2 + k * 2) -48k + -8k2 = -5k + (-8 + 2k) Reorder the terms: -48k + -8k2 = -8 + -5k + 2k Combine like terms: -5k + 2k = -3k -48k + -8k2 = -8 + -3k Solving -48k + -8k2 = -8 + -3k Solving for variable 'k'. Reorder the terms: 8 + -48k + 3k + -8k2 = -8 + -3k + 8 + 3k Combine like terms: -48k + 3k = -45k 8 + -45k + -8k2 = -8 + -3k + 8 + 3k Reorder the terms: 8 + -45k + -8k2 = -8 + 8 + -3k + 3k Combine like terms: -8 + 8 = 0 8 + -45k + -8k2 = 0 + -3k + 3k 8 + -45k + -8k2 = -3k + 3k Combine like terms: -3k + 3k = 0 8 + -45k + -8k2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1 + 5.625k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 5.625k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 5.625k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 5.625k + k2 = 0 + 1 5.625k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 5.625k + k2 = 1 The k term is 5.625k. Take half its coefficient (2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. 5.625k + 7.91015625 + k2 = 1 + 7.91015625 Reorder the terms: 7.91015625 + 5.625k + k2 = 1 + 7.91015625 Combine like terms: 1 + 7.91015625 = 8.91015625 7.91015625 + 5.625k + k2 = 8.91015625 Factor a perfect square on the left side: (k + 2.8125)(k + 2.8125) = 8.91015625 Calculate the square root of the right side: 2.984988484 Break this problem into two subproblems by setting (k + 2.8125) equal to 2.984988484 and -2.984988484.Subproblem 1
k + 2.8125 = 2.984988484 Simplifying k + 2.8125 = 2.984988484 Reorder the terms: 2.8125 + k = 2.984988484 Solving 2.8125 + k = 2.984988484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + k = 2.984988484 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + k = 2.984988484 + -2.8125 k = 2.984988484 + -2.8125 Combine like terms: 2.984988484 + -2.8125 = 0.172488484 k = 0.172488484 Simplifying k = 0.172488484Subproblem 2
k + 2.8125 = -2.984988484 Simplifying k + 2.8125 = -2.984988484 Reorder the terms: 2.8125 + k = -2.984988484 Solving 2.8125 + k = -2.984988484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + k = -2.984988484 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + k = -2.984988484 + -2.8125 k = -2.984988484 + -2.8125 Combine like terms: -2.984988484 + -2.8125 = -5.797488484 k = -5.797488484 Simplifying k = -5.797488484Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.172488484, -5.797488484}
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