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Simplifying 3000 = L(L + -8) Reorder the terms: 3000 = L(-8 + L) 3000 = (-8 * L + L * L) 3000 = (-8L + L2) Solving 3000 = -8L + L2 Solving for variable 'L'. Reorder the terms: 3000 + 8L + -1L2 = -8L + 8L + L2 + -1L2 Combine like terms: -8L + 8L = 0 3000 + 8L + -1L2 = 0 + L2 + -1L2 3000 + 8L + -1L2 = L2 + -1L2 Combine like terms: L2 + -1L2 = 0 3000 + 8L + -1L2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -3000 + -8L + L2 = 0 Move the constant term to the right: Add '3000' to each side of the equation. -3000 + -8L + 3000 + L2 = 0 + 3000 Reorder the terms: -3000 + 3000 + -8L + L2 = 0 + 3000 Combine like terms: -3000 + 3000 = 0 0 + -8L + L2 = 0 + 3000 -8L + L2 = 0 + 3000 Combine like terms: 0 + 3000 = 3000 -8L + L2 = 3000 The L term is -8L. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8L + 16 + L2 = 3000 + 16 Reorder the terms: 16 + -8L + L2 = 3000 + 16 Combine like terms: 3000 + 16 = 3016 16 + -8L + L2 = 3016 Factor a perfect square on the left side: (L + -4)(L + -4) = 3016 Calculate the square root of the right side: 54.918120871 Break this problem into two subproblems by setting (L + -4) equal to 54.918120871 and -54.918120871.Subproblem 1
L + -4 = 54.918120871 Simplifying L + -4 = 54.918120871 Reorder the terms: -4 + L = 54.918120871 Solving -4 + L = 54.918120871 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + L = 54.918120871 + 4 Combine like terms: -4 + 4 = 0 0 + L = 54.918120871 + 4 L = 54.918120871 + 4 Combine like terms: 54.918120871 + 4 = 58.918120871 L = 58.918120871 Simplifying L = 58.918120871Subproblem 2
L + -4 = -54.918120871 Simplifying L + -4 = -54.918120871 Reorder the terms: -4 + L = -54.918120871 Solving -4 + L = -54.918120871 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + L = -54.918120871 + 4 Combine like terms: -4 + 4 = 0 0 + L = -54.918120871 + 4 L = -54.918120871 + 4 Combine like terms: -54.918120871 + 4 = -50.918120871 L = -50.918120871 Simplifying L = -50.918120871Solution
The solution to the problem is based on the solutions from the subproblems. L = {58.918120871, -50.918120871}
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