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Simplifying L2 + -23L = 90 Reorder the terms: -23L + L2 = 90 Solving -23L + L2 = 90 Solving for variable 'L'. Reorder the terms: -90 + -23L + L2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + -23L + L2 = 0 Begin completing the square. Move the constant term to the right: Add '90' to each side of the equation. -90 + -23L + 90 + L2 = 0 + 90 Reorder the terms: -90 + 90 + -23L + L2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + -23L + L2 = 0 + 90 -23L + L2 = 0 + 90 Combine like terms: 0 + 90 = 90 -23L + L2 = 90 The L term is -23L. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23L + 132.25 + L2 = 90 + 132.25 Reorder the terms: 132.25 + -23L + L2 = 90 + 132.25 Combine like terms: 90 + 132.25 = 222.25 132.25 + -23L + L2 = 222.25 Factor a perfect square on the left side: (L + -11.5)(L + -11.5) = 222.25 Calculate the square root of the right side: 14.908051516 Break this problem into two subproblems by setting (L + -11.5) equal to 14.908051516 and -14.908051516.Subproblem 1
L + -11.5 = 14.908051516 Simplifying L + -11.5 = 14.908051516 Reorder the terms: -11.5 + L = 14.908051516 Solving -11.5 + L = 14.908051516 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + L = 14.908051516 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + L = 14.908051516 + 11.5 L = 14.908051516 + 11.5 Combine like terms: 14.908051516 + 11.5 = 26.408051516 L = 26.408051516 Simplifying L = 26.408051516Subproblem 2
L + -11.5 = -14.908051516 Simplifying L + -11.5 = -14.908051516 Reorder the terms: -11.5 + L = -14.908051516 Solving -11.5 + L = -14.908051516 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + L = -14.908051516 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + L = -14.908051516 + 11.5 L = -14.908051516 + 11.5 Combine like terms: -14.908051516 + 11.5 = -3.408051516 L = -3.408051516 Simplifying L = -3.408051516Solution
The solution to the problem is based on the solutions from the subproblems. L = {26.408051516, -3.408051516}
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