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Simplifying (50 + 12L + -1.5L2) * 10 = 10 Reorder the terms for easier multiplication: 10(50 + 12L + -1.5L2) = 10 (50 * 10 + 12L * 10 + -1.5L2 * 10) = 10 (500 + 120L + -15L2) = 10 Solving 500 + 120L + -15L2 = 10 Solving for variable 'L'. Reorder the terms: 500 + -10 + 120L + -15L2 = 10 + -10 Combine like terms: 500 + -10 = 490 490 + 120L + -15L2 = 10 + -10 Combine like terms: 10 + -10 = 0 490 + 120L + -15L2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(98 + 24L + -3L2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(98 + 24L + -3L2)' equal to zero and attempt to solve: Simplifying 98 + 24L + -3L2 = 0 Solving 98 + 24L + -3L2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -32.66666667 + -8L + L2 = 0 Move the constant term to the right: Add '32.66666667' to each side of the equation. -32.66666667 + -8L + 32.66666667 + L2 = 0 + 32.66666667 Reorder the terms: -32.66666667 + 32.66666667 + -8L + L2 = 0 + 32.66666667 Combine like terms: -32.66666667 + 32.66666667 = 0.00000000 0.00000000 + -8L + L2 = 0 + 32.66666667 -8L + L2 = 0 + 32.66666667 Combine like terms: 0 + 32.66666667 = 32.66666667 -8L + L2 = 32.66666667 The L term is -8L. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8L + 16 + L2 = 32.66666667 + 16 Reorder the terms: 16 + -8L + L2 = 32.66666667 + 16 Combine like terms: 32.66666667 + 16 = 48.66666667 16 + -8L + L2 = 48.66666667 Factor a perfect square on the left side: (L + -4)(L + -4) = 48.66666667 Calculate the square root of the right side: 6.976149846 Break this problem into two subproblems by setting (L + -4) equal to 6.976149846 and -6.976149846.Subproblem 1
L + -4 = 6.976149846 Simplifying L + -4 = 6.976149846 Reorder the terms: -4 + L = 6.976149846 Solving -4 + L = 6.976149846 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + L = 6.976149846 + 4 Combine like terms: -4 + 4 = 0 0 + L = 6.976149846 + 4 L = 6.976149846 + 4 Combine like terms: 6.976149846 + 4 = 10.976149846 L = 10.976149846 Simplifying L = 10.976149846Subproblem 2
L + -4 = -6.976149846 Simplifying L + -4 = -6.976149846 Reorder the terms: -4 + L = -6.976149846 Solving -4 + L = -6.976149846 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + L = -6.976149846 + 4 Combine like terms: -4 + 4 = 0 0 + L = -6.976149846 + 4 L = -6.976149846 + 4 Combine like terms: -6.976149846 + 4 = -2.976149846 L = -2.976149846 Simplifying L = -2.976149846Solution
The solution to the problem is based on the solutions from the subproblems. L = {10.976149846, -2.976149846}Solution
L = {10.976149846, -2.976149846}
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