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Simplifying (-12 + 3l + (-4l + l2)) = 12 Remove parenthesis around (-4l + l2) (-12 + 3l + -4l + l2) = 12 Combine like terms: 3l + -4l = -1l (-12 + -1l + l2) = 12 Remove parenthesis around (-12 + -1l + l2) -12 + -1l + l2 = 12 Solving -12 + -1l + l2 = 12 Solving for variable 'l'. Reorder the terms: -12 + -12 + -1l + l2 = 12 + -12 Combine like terms: -12 + -12 = -24 -24 + -1l + l2 = 12 + -12 Combine like terms: 12 + -12 = 0 -24 + -1l + l2 = 0 Begin completing the square. Move the constant term to the right: Add '24' to each side of the equation. -24 + -1l + 24 + l2 = 0 + 24 Reorder the terms: -24 + 24 + -1l + l2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -1l + l2 = 0 + 24 -1l + l2 = 0 + 24 Combine like terms: 0 + 24 = 24 -1l + l2 = 24 The l term is -1l. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1l + 0.25 + l2 = 24 + 0.25 Reorder the terms: 0.25 + -1l + l2 = 24 + 0.25 Combine like terms: 24 + 0.25 = 24.25 0.25 + -1l + l2 = 24.25 Factor a perfect square on the left side: (l + -0.5)(l + -0.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (l + -0.5) equal to 4.924428901 and -4.924428901.Subproblem 1
l + -0.5 = 4.924428901 Simplifying l + -0.5 = 4.924428901 Reorder the terms: -0.5 + l = 4.924428901 Solving -0.5 + l = 4.924428901 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + l = 4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + l = 4.924428901 + 0.5 l = 4.924428901 + 0.5 Combine like terms: 4.924428901 + 0.5 = 5.424428901 l = 5.424428901 Simplifying l = 5.424428901Subproblem 2
l + -0.5 = -4.924428901 Simplifying l + -0.5 = -4.924428901 Reorder the terms: -0.5 + l = -4.924428901 Solving -0.5 + l = -4.924428901 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + l = -4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + l = -4.924428901 + 0.5 l = -4.924428901 + 0.5 Combine like terms: -4.924428901 + 0.5 = -4.424428901 l = -4.424428901 Simplifying l = -4.424428901Solution
The solution to the problem is based on the solutions from the subproblems. l = {5.424428901, -4.424428901}
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