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Simplifying 54.3(0.5 + -1x)(0.4 + -1x) = 0 Multiply (0.5 + -1x) * (0.4 + -1x) 54.3(0.5(0.4 + -1x) + -1x * (0.4 + -1x)) = 0 54.3((0.4 * 0.5 + -1x * 0.5) + -1x * (0.4 + -1x)) = 0 54.3((0.2 + -0.5x) + -1x * (0.4 + -1x)) = 0 54.3(0.2 + -0.5x + (0.4 * -1x + -1x * -1x)) = 0 54.3(0.2 + -0.5x + (-0.4x + 1x2)) = 0 Combine like terms: -0.5x + -0.4x = -0.9x 54.3(0.2 + -0.9x + 1x2) = 0 (0.2 * 54.3 + -0.9x * 54.3 + 1x2 * 54.3) = 0 (10.86 + -48.87x + 54.3x2) = 0 Solving 10.86 + -48.87x + 54.3x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 54.3 the coefficient of the squared term: Divide each side by '54.3'. 0.2 + -0.9x + x2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -0.9x + -0.2 + x2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -0.9x + x2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -0.9x + x2 = 0 + -0.2 -0.9x + x2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -0.9x + x2 = -0.2 The x term is -0.9x. Take half its coefficient (-0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. -0.9x + 0.2025 + x2 = -0.2 + 0.2025 Reorder the terms: 0.2025 + -0.9x + x2 = -0.2 + 0.2025 Combine like terms: -0.2 + 0.2025 = 0.0025 0.2025 + -0.9x + x2 = 0.0025 Factor a perfect square on the left side: (x + -0.45)(x + -0.45) = 0.0025 Calculate the square root of the right side: 0.05 Break this problem into two subproblems by setting (x + -0.45) equal to 0.05 and -0.05.Subproblem 1
x + -0.45 = 0.05 Simplifying x + -0.45 = 0.05 Reorder the terms: -0.45 + x = 0.05 Solving -0.45 + x = 0.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + x = 0.05 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + x = 0.05 + 0.45 x = 0.05 + 0.45 Combine like terms: 0.05 + 0.45 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + -0.45 = -0.05 Simplifying x + -0.45 = -0.05 Reorder the terms: -0.45 + x = -0.05 Solving -0.45 + x = -0.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + x = -0.05 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + x = -0.05 + 0.45 x = -0.05 + 0.45 Combine like terms: -0.05 + 0.45 = 0.4 x = 0.4 Simplifying x = 0.4Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, 0.4}
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