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Simplifying 42L + 18L2 + -2L3 = 0 Solving 42L + 18L2 + -2L3 = 0 Solving for variable 'L'. Factor out the Greatest Common Factor (GCF), '2L'. 2L(21 + 9L + -1L2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'L' equal to zero and attempt to solve: Simplifying L = 0 Solving L = 0 Move all terms containing L to the left, all other terms to the right. Simplifying L = 0Subproblem 2
Set the factor '(21 + 9L + -1L2)' equal to zero and attempt to solve: Simplifying 21 + 9L + -1L2 = 0 Solving 21 + 9L + -1L2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -21 + -9L + L2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + -9L + 21 + L2 = 0 + 21 Reorder the terms: -21 + 21 + -9L + L2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + -9L + L2 = 0 + 21 -9L + L2 = 0 + 21 Combine like terms: 0 + 21 = 21 -9L + L2 = 21 The L term is -9L. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9L + 20.25 + L2 = 21 + 20.25 Reorder the terms: 20.25 + -9L + L2 = 21 + 20.25 Combine like terms: 21 + 20.25 = 41.25 20.25 + -9L + L2 = 41.25 Factor a perfect square on the left side: (L + -4.5)(L + -4.5) = 41.25 Calculate the square root of the right side: 6.422616289 Break this problem into two subproblems by setting (L + -4.5) equal to 6.422616289 and -6.422616289.Subproblem 1
L + -4.5 = 6.422616289 Simplifying L + -4.5 = 6.422616289 Reorder the terms: -4.5 + L = 6.422616289 Solving -4.5 + L = 6.422616289 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + L = 6.422616289 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + L = 6.422616289 + 4.5 L = 6.422616289 + 4.5 Combine like terms: 6.422616289 + 4.5 = 10.922616289 L = 10.922616289 Simplifying L = 10.922616289Subproblem 2
L + -4.5 = -6.422616289 Simplifying L + -4.5 = -6.422616289 Reorder the terms: -4.5 + L = -6.422616289 Solving -4.5 + L = -6.422616289 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + L = -6.422616289 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + L = -6.422616289 + 4.5 L = -6.422616289 + 4.5 Combine like terms: -6.422616289 + 4.5 = -1.922616289 L = -1.922616289 Simplifying L = -1.922616289Solution
The solution to the problem is based on the solutions from the subproblems. L = {10.922616289, -1.922616289}Solution
L = {0, 10.922616289, -1.922616289}
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