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Simplifying 42 + 36L + -6L2 = 94.5 Solving 42 + 36L + -6L2 = 94.5 Solving for variable 'L'. Reorder the terms: 42 + -94.5 + 36L + -6L2 = 94.5 + -94.5 Combine like terms: 42 + -94.5 = -52.5 -52.5 + 36L + -6L2 = 94.5 + -94.5 Combine like terms: 94.5 + -94.5 = 0.0 -52.5 + 36L + -6L2 = 0.0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 8.75 + -6L + L2 = 0.0 Move the constant term to the right: Add '-8.75' to each side of the equation. 8.75 + -6L + -8.75 + L2 = 0.0 + -8.75 Reorder the terms: 8.75 + -8.75 + -6L + L2 = 0.0 + -8.75 Combine like terms: 8.75 + -8.75 = 0.00 0.00 + -6L + L2 = 0.0 + -8.75 -6L + L2 = 0.0 + -8.75 Combine like terms: 0.0 + -8.75 = -8.75 -6L + L2 = -8.75 The L term is -6L. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6L + 9 + L2 = -8.75 + 9 Reorder the terms: 9 + -6L + L2 = -8.75 + 9 Combine like terms: -8.75 + 9 = 0.25 9 + -6L + L2 = 0.25 Factor a perfect square on the left side: (L + -3)(L + -3) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (L + -3) equal to 0.5 and -0.5.Subproblem 1
L + -3 = 0.5 Simplifying L + -3 = 0.5 Reorder the terms: -3 + L = 0.5 Solving -3 + L = 0.5 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + L = 0.5 + 3 Combine like terms: -3 + 3 = 0 0 + L = 0.5 + 3 L = 0.5 + 3 Combine like terms: 0.5 + 3 = 3.5 L = 3.5 Simplifying L = 3.5Subproblem 2
L + -3 = -0.5 Simplifying L + -3 = -0.5 Reorder the terms: -3 + L = -0.5 Solving -3 + L = -0.5 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + L = -0.5 + 3 Combine like terms: -3 + 3 = 0 0 + L = -0.5 + 3 L = -0.5 + 3 Combine like terms: -0.5 + 3 = 2.5 L = 2.5 Simplifying L = 2.5Solution
The solution to the problem is based on the solutions from the subproblems. L = {3.5, 2.5}
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