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Simplifying l2 + 3l + -75 = 0 Reorder the terms: -75 + 3l + l2 = 0 Solving -75 + 3l + l2 = 0 Solving for variable 'l'. Begin completing the square. Move the constant term to the right: Add '75' to each side of the equation. -75 + 3l + 75 + l2 = 0 + 75 Reorder the terms: -75 + 75 + 3l + l2 = 0 + 75 Combine like terms: -75 + 75 = 0 0 + 3l + l2 = 0 + 75 3l + l2 = 0 + 75 Combine like terms: 0 + 75 = 75 3l + l2 = 75 The l term is 3l. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3l + 2.25 + l2 = 75 + 2.25 Reorder the terms: 2.25 + 3l + l2 = 75 + 2.25 Combine like terms: 75 + 2.25 = 77.25 2.25 + 3l + l2 = 77.25 Factor a perfect square on the left side: (l + 1.5)(l + 1.5) = 77.25 Calculate the square root of the right side: 8.789197916 Break this problem into two subproblems by setting (l + 1.5) equal to 8.789197916 and -8.789197916.Subproblem 1
l + 1.5 = 8.789197916 Simplifying l + 1.5 = 8.789197916 Reorder the terms: 1.5 + l = 8.789197916 Solving 1.5 + l = 8.789197916 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + l = 8.789197916 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + l = 8.789197916 + -1.5 l = 8.789197916 + -1.5 Combine like terms: 8.789197916 + -1.5 = 7.289197916 l = 7.289197916 Simplifying l = 7.289197916Subproblem 2
l + 1.5 = -8.789197916 Simplifying l + 1.5 = -8.789197916 Reorder the terms: 1.5 + l = -8.789197916 Solving 1.5 + l = -8.789197916 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + l = -8.789197916 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + l = -8.789197916 + -1.5 l = -8.789197916 + -1.5 Combine like terms: -8.789197916 + -1.5 = -10.289197916 l = -10.289197916 Simplifying l = -10.289197916Solution
The solution to the problem is based on the solutions from the subproblems. l = {7.289197916, -10.289197916}
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