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Simplifying 0.52(0.50 + -1x)(0.045 + -1x) = (0.050 + x)(0.040 + x) Multiply (0.50 + -1x) * (0.045 + -1x) 0.52(0.50(0.045 + -1x) + -1x * (0.045 + -1x)) = (0.050 + x)(0.040 + x) 0.52((0.045 * 0.50 + -1x * 0.50) + -1x * (0.045 + -1x)) = (0.050 + x)(0.040 + x) 0.52((0.0225 + -0.5x) + -1x * (0.045 + -1x)) = (0.050 + x)(0.040 + x) 0.52(0.0225 + -0.5x + (0.045 * -1x + -1x * -1x)) = (0.050 + x)(0.040 + x) 0.52(0.0225 + -0.5x + (-0.045x + 1x2)) = (0.050 + x)(0.040 + x) Combine like terms: -0.5x + -0.045x = -0.545x 0.52(0.0225 + -0.545x + 1x2) = (0.050 + x)(0.040 + x) (0.0225 * 0.52 + -0.545x * 0.52 + 1x2 * 0.52) = (0.050 + x)(0.040 + x) (0.0117 + -0.2834x + 0.52x2) = (0.050 + x)(0.040 + x) Multiply (0.050 + x) * (0.040 + x) 0.0117 + -0.2834x + 0.52x2 = (0.050(0.040 + x) + x(0.040 + x)) 0.0117 + -0.2834x + 0.52x2 = ((0.040 * 0.050 + x * 0.050) + x(0.040 + x)) 0.0117 + -0.2834x + 0.52x2 = ((0.002 + 0.050x) + x(0.040 + x)) 0.0117 + -0.2834x + 0.52x2 = (0.002 + 0.050x + (0.040 * x + x * x)) 0.0117 + -0.2834x + 0.52x2 = (0.002 + 0.050x + (0.040x + x2)) Combine like terms: 0.050x + 0.040x = 0.09x 0.0117 + -0.2834x + 0.52x2 = (0.002 + 0.09x + x2) Solving 0.0117 + -0.2834x + 0.52x2 = 0.002 + 0.09x + x2 Solving for variable 'x'. Reorder the terms: 0.0117 + -0.002 + -0.2834x + -0.09x + 0.52x2 + -1x2 = 0.002 + 0.09x + x2 + -0.002 + -0.09x + -1x2 Combine like terms: 0.0117 + -0.002 = 0.0097 0.0097 + -0.2834x + -0.09x + 0.52x2 + -1x2 = 0.002 + 0.09x + x2 + -0.002 + -0.09x + -1x2 Combine like terms: -0.2834x + -0.09x = -0.3734x 0.0097 + -0.3734x + 0.52x2 + -1x2 = 0.002 + 0.09x + x2 + -0.002 + -0.09x + -1x2 Combine like terms: 0.52x2 + -1x2 = -0.48x2 0.0097 + -0.3734x + -0.48x2 = 0.002 + 0.09x + x2 + -0.002 + -0.09x + -1x2 Reorder the terms: 0.0097 + -0.3734x + -0.48x2 = 0.002 + -0.002 + 0.09x + -0.09x + x2 + -1x2 Combine like terms: 0.002 + -0.002 = 0.000 0.0097 + -0.3734x + -0.48x2 = 0.000 + 0.09x + -0.09x + x2 + -1x2 0.0097 + -0.3734x + -0.48x2 = 0.09x + -0.09x + x2 + -1x2 Combine like terms: 0.09x + -0.09x = 0.00 0.0097 + -0.3734x + -0.48x2 = 0.00 + x2 + -1x2 0.0097 + -0.3734x + -0.48x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 0.0097 + -0.3734x + -0.48x2 = 0 Begin completing the square. Divide all terms by -0.48 the coefficient of the squared term: Divide each side by '-0.48'. -0.02020833333 + 0.7779166667x + x2 = 0 Move the constant term to the right: Add '0.02020833333' to each side of the equation. -0.02020833333 + 0.7779166667x + 0.02020833333 + x2 = 0 + 0.02020833333 Reorder the terms: -0.02020833333 + 0.02020833333 + 0.7779166667x + x2 = 0 + 0.02020833333 Combine like terms: -0.02020833333 + 0.02020833333 = 0.00000000000 0.00000000000 + 0.7779166667x + x2 = 0 + 0.02020833333 0.7779166667x + x2 = 0 + 0.02020833333 Combine like terms: 0 + 0.02020833333 = 0.02020833333 0.7779166667x + x2 = 0.02020833333 The x term is 0.7779166667x. Take half its coefficient (0.3889583334). Square it (0.1512885851) and add it to both sides. Add '0.1512885851' to each side of the equation. 0.7779166667x + 0.1512885851 + x2 = 0.02020833333 + 0.1512885851 Reorder the terms: 0.1512885851 + 0.7779166667x + x2 = 0.02020833333 + 0.1512885851 Combine like terms: 0.02020833333 + 0.1512885851 = 0.17149691843 0.1512885851 + 0.7779166667x + x2 = 0.17149691843 Factor a perfect square on the left side: (x + 0.3889583334)(x + 0.3889583334) = 0.17149691843 Calculate the square root of the right side: 0.414121864 Break this problem into two subproblems by setting (x + 0.3889583334) equal to 0.414121864 and -0.414121864.Subproblem 1
x + 0.3889583334 = 0.414121864 Simplifying x + 0.3889583334 = 0.414121864 Reorder the terms: 0.3889583334 + x = 0.414121864 Solving 0.3889583334 + x = 0.414121864 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3889583334' to each side of the equation. 0.3889583334 + -0.3889583334 + x = 0.414121864 + -0.3889583334 Combine like terms: 0.3889583334 + -0.3889583334 = 0.0000000000 0.0000000000 + x = 0.414121864 + -0.3889583334 x = 0.414121864 + -0.3889583334 Combine like terms: 0.414121864 + -0.3889583334 = 0.0251635306 x = 0.0251635306 Simplifying x = 0.0251635306Subproblem 2
x + 0.3889583334 = -0.414121864 Simplifying x + 0.3889583334 = -0.414121864 Reorder the terms: 0.3889583334 + x = -0.414121864 Solving 0.3889583334 + x = -0.414121864 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3889583334' to each side of the equation. 0.3889583334 + -0.3889583334 + x = -0.414121864 + -0.3889583334 Combine like terms: 0.3889583334 + -0.3889583334 = 0.0000000000 0.0000000000 + x = -0.414121864 + -0.3889583334 x = -0.414121864 + -0.3889583334 Combine like terms: -0.414121864 + -0.3889583334 = -0.8030801974 x = -0.8030801974 Simplifying x = -0.8030801974Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.0251635306, -0.8030801974}
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