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Simplifying (80 + -1x) * 2x = 108 Reorder the terms for easier multiplication: 2x(80 + -1x) = 108 (80 * 2x + -1x * 2x) = 108 (160x + -2x2) = 108 Solving 160x + -2x2 = 108 Solving for variable 'x'. Reorder the terms: -108 + 160x + -2x2 = 108 + -108 Combine like terms: 108 + -108 = 0 -108 + 160x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-54 + 80x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-54 + 80x + -1x2)' equal to zero and attempt to solve: Simplifying -54 + 80x + -1x2 = 0 Solving -54 + 80x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 54 + -80x + x2 = 0 Move the constant term to the right: Add '-54' to each side of the equation. 54 + -80x + -54 + x2 = 0 + -54 Reorder the terms: 54 + -54 + -80x + x2 = 0 + -54 Combine like terms: 54 + -54 = 0 0 + -80x + x2 = 0 + -54 -80x + x2 = 0 + -54 Combine like terms: 0 + -54 = -54 -80x + x2 = -54 The x term is -80x. Take half its coefficient (-40). Square it (1600) and add it to both sides. Add '1600' to each side of the equation. -80x + 1600 + x2 = -54 + 1600 Reorder the terms: 1600 + -80x + x2 = -54 + 1600 Combine like terms: -54 + 1600 = 1546 1600 + -80x + x2 = 1546 Factor a perfect square on the left side: (x + -40)(x + -40) = 1546 Calculate the square root of the right side: 39.319206503 Break this problem into two subproblems by setting (x + -40) equal to 39.319206503 and -39.319206503.Subproblem 1
x + -40 = 39.319206503 Simplifying x + -40 = 39.319206503 Reorder the terms: -40 + x = 39.319206503 Solving -40 + x = 39.319206503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = 39.319206503 + 40 Combine like terms: -40 + 40 = 0 0 + x = 39.319206503 + 40 x = 39.319206503 + 40 Combine like terms: 39.319206503 + 40 = 79.319206503 x = 79.319206503 Simplifying x = 79.319206503Subproblem 2
x + -40 = -39.319206503 Simplifying x + -40 = -39.319206503 Reorder the terms: -40 + x = -39.319206503 Solving -40 + x = -39.319206503 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = -39.319206503 + 40 Combine like terms: -40 + 40 = 0 0 + x = -39.319206503 + 40 x = -39.319206503 + 40 Combine like terms: -39.319206503 + 40 = 0.680793497 x = 0.680793497 Simplifying x = 0.680793497Solution
The solution to the problem is based on the solutions from the subproblems. x = {79.319206503, 0.680793497}Solution
x = {79.319206503, 0.680793497}
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