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Simplifying 24x + 30x + 4x2 = 378 Combine like terms: 24x + 30x = 54x 54x + 4x2 = 378 Solving 54x + 4x2 = 378 Solving for variable 'x'. Reorder the terms: -378 + 54x + 4x2 = 378 + -378 Combine like terms: 378 + -378 = 0 -378 + 54x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-189 + 27x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-189 + 27x + 2x2)' equal to zero and attempt to solve: Simplifying -189 + 27x + 2x2 = 0 Solving -189 + 27x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -94.5 + 13.5x + x2 = 0 Move the constant term to the right: Add '94.5' to each side of the equation. -94.5 + 13.5x + 94.5 + x2 = 0 + 94.5 Reorder the terms: -94.5 + 94.5 + 13.5x + x2 = 0 + 94.5 Combine like terms: -94.5 + 94.5 = 0.0 0.0 + 13.5x + x2 = 0 + 94.5 13.5x + x2 = 0 + 94.5 Combine like terms: 0 + 94.5 = 94.5 13.5x + x2 = 94.5 The x term is 13.5x. Take half its coefficient (6.75). Square it (45.5625) and add it to both sides. Add '45.5625' to each side of the equation. 13.5x + 45.5625 + x2 = 94.5 + 45.5625 Reorder the terms: 45.5625 + 13.5x + x2 = 94.5 + 45.5625 Combine like terms: 94.5 + 45.5625 = 140.0625 45.5625 + 13.5x + x2 = 140.0625 Factor a perfect square on the left side: (x + 6.75)(x + 6.75) = 140.0625 Calculate the square root of the right side: 11.834800379 Break this problem into two subproblems by setting (x + 6.75) equal to 11.834800379 and -11.834800379.Subproblem 1
x + 6.75 = 11.834800379 Simplifying x + 6.75 = 11.834800379 Reorder the terms: 6.75 + x = 11.834800379 Solving 6.75 + x = 11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + x = 11.834800379 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + x = 11.834800379 + -6.75 x = 11.834800379 + -6.75 Combine like terms: 11.834800379 + -6.75 = 5.084800379 x = 5.084800379 Simplifying x = 5.084800379Subproblem 2
x + 6.75 = -11.834800379 Simplifying x + 6.75 = -11.834800379 Reorder the terms: 6.75 + x = -11.834800379 Solving 6.75 + x = -11.834800379 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + x = -11.834800379 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + x = -11.834800379 + -6.75 x = -11.834800379 + -6.75 Combine like terms: -11.834800379 + -6.75 = -18.584800379 x = -18.584800379 Simplifying x = -18.584800379Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.084800379, -18.584800379}Solution
x = {5.084800379, -18.584800379}
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