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Simplifying L(5 + 2L) = 46 (5 * L + 2L * L) = 46 (5L + 2L2) = 46 Solving 5L + 2L2 = 46 Solving for variable 'L'. Reorder the terms: -46 + 5L + 2L2 = 46 + -46 Combine like terms: 46 + -46 = 0 -46 + 5L + 2L2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -23 + 2.5L + L2 = 0 Move the constant term to the right: Add '23' to each side of the equation. -23 + 2.5L + 23 + L2 = 0 + 23 Reorder the terms: -23 + 23 + 2.5L + L2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 2.5L + L2 = 0 + 23 2.5L + L2 = 0 + 23 Combine like terms: 0 + 23 = 23 2.5L + L2 = 23 The L term is 2.5L. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5L + 1.5625 + L2 = 23 + 1.5625 Reorder the terms: 1.5625 + 2.5L + L2 = 23 + 1.5625 Combine like terms: 23 + 1.5625 = 24.5625 1.5625 + 2.5L + L2 = 24.5625 Factor a perfect square on the left side: (L + 1.25)(L + 1.25) = 24.5625 Calculate the square root of the right side: 4.9560569 Break this problem into two subproblems by setting (L + 1.25) equal to 4.9560569 and -4.9560569.Subproblem 1
L + 1.25 = 4.9560569 Simplifying L + 1.25 = 4.9560569 Reorder the terms: 1.25 + L = 4.9560569 Solving 1.25 + L = 4.9560569 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + L = 4.9560569 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + L = 4.9560569 + -1.25 L = 4.9560569 + -1.25 Combine like terms: 4.9560569 + -1.25 = 3.7060569 L = 3.7060569 Simplifying L = 3.7060569Subproblem 2
L + 1.25 = -4.9560569 Simplifying L + 1.25 = -4.9560569 Reorder the terms: 1.25 + L = -4.9560569 Solving 1.25 + L = -4.9560569 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + L = -4.9560569 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + L = -4.9560569 + -1.25 L = -4.9560569 + -1.25 Combine like terms: -4.9560569 + -1.25 = -6.2060569 L = -6.2060569 Simplifying L = -6.2060569Solution
The solution to the problem is based on the solutions from the subproblems. L = {3.7060569, -6.2060569}
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