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Simplifying L(2L + 1) = 32 Reorder the terms: L(1 + 2L) = 32 (1 * L + 2L * L) = 32 (1L + 2L2) = 32 Solving 1L + 2L2 = 32 Solving for variable 'L'. Reorder the terms: -32 + 1L + 2L2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 1L + 2L2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -16 + 0.5L + L2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 0.5L + 16 + L2 = 0 + 16 Reorder the terms: -16 + 16 + 0.5L + L2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 0.5L + L2 = 0 + 16 0.5L + L2 = 0 + 16 Combine like terms: 0 + 16 = 16 0.5L + L2 = 16 The L term is 0.5L. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5L + 0.0625 + L2 = 16 + 0.0625 Reorder the terms: 0.0625 + 0.5L + L2 = 16 + 0.0625 Combine like terms: 16 + 0.0625 = 16.0625 0.0625 + 0.5L + L2 = 16.0625 Factor a perfect square on the left side: (L + 0.25)(L + 0.25) = 16.0625 Calculate the square root of the right side: 4.007804885 Break this problem into two subproblems by setting (L + 0.25) equal to 4.007804885 and -4.007804885.Subproblem 1
L + 0.25 = 4.007804885 Simplifying L + 0.25 = 4.007804885 Reorder the terms: 0.25 + L = 4.007804885 Solving 0.25 + L = 4.007804885 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + L = 4.007804885 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + L = 4.007804885 + -0.25 L = 4.007804885 + -0.25 Combine like terms: 4.007804885 + -0.25 = 3.757804885 L = 3.757804885 Simplifying L = 3.757804885Subproblem 2
L + 0.25 = -4.007804885 Simplifying L + 0.25 = -4.007804885 Reorder the terms: 0.25 + L = -4.007804885 Solving 0.25 + L = -4.007804885 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + L = -4.007804885 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + L = -4.007804885 + -0.25 L = -4.007804885 + -0.25 Combine like terms: -4.007804885 + -0.25 = -4.257804885 L = -4.257804885 Simplifying L = -4.257804885Solution
The solution to the problem is based on the solutions from the subproblems. L = {3.757804885, -4.257804885}
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