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Simplifying (23 + -2x)(13 + -2x)(x) = x Reorder the terms for easier multiplication: x(23 + -2x)(13 + -2x) = x Multiply (23 + -2x) * (13 + -2x) x(23(13 + -2x) + -2x * (13 + -2x)) = x x((13 * 23 + -2x * 23) + -2x * (13 + -2x)) = x x((299 + -46x) + -2x * (13 + -2x)) = x x(299 + -46x + (13 * -2x + -2x * -2x)) = x x(299 + -46x + (-26x + 4x2)) = x Combine like terms: -46x + -26x = -72x x(299 + -72x + 4x2) = x (299 * x + -72x * x + 4x2 * x) = x (299x + -72x2 + 4x3) = x Solving 299x + -72x2 + 4x3 = x Solving for variable 'x'. Reorder the terms: 299x + -1x + -72x2 + 4x3 = x + -1x Combine like terms: 299x + -1x = 298x 298x + -72x2 + 4x3 = x + -1x Combine like terms: x + -1x = 0 298x + -72x2 + 4x3 = 0 Factor out the Greatest Common Factor (GCF), '2x'. 2x(149 + -36x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(149 + -36x + 2x2)' equal to zero and attempt to solve: Simplifying 149 + -36x + 2x2 = 0 Solving 149 + -36x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 74.5 + -18x + x2 = 0 Move the constant term to the right: Add '-74.5' to each side of the equation. 74.5 + -18x + -74.5 + x2 = 0 + -74.5 Reorder the terms: 74.5 + -74.5 + -18x + x2 = 0 + -74.5 Combine like terms: 74.5 + -74.5 = 0.0 0.0 + -18x + x2 = 0 + -74.5 -18x + x2 = 0 + -74.5 Combine like terms: 0 + -74.5 = -74.5 -18x + x2 = -74.5 The x term is -18x. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18x + 81 + x2 = -74.5 + 81 Reorder the terms: 81 + -18x + x2 = -74.5 + 81 Combine like terms: -74.5 + 81 = 6.5 81 + -18x + x2 = 6.5 Factor a perfect square on the left side: (x + -9)(x + -9) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (x + -9) equal to 2.549509757 and -2.549509757.Subproblem 1
x + -9 = 2.549509757 Simplifying x + -9 = 2.549509757 Reorder the terms: -9 + x = 2.549509757 Solving -9 + x = 2.549509757 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = 2.549509757 + 9 Combine like terms: -9 + 9 = 0 0 + x = 2.549509757 + 9 x = 2.549509757 + 9 Combine like terms: 2.549509757 + 9 = 11.549509757 x = 11.549509757 Simplifying x = 11.549509757Subproblem 2
x + -9 = -2.549509757 Simplifying x + -9 = -2.549509757 Reorder the terms: -9 + x = -2.549509757 Solving -9 + x = -2.549509757 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = -2.549509757 + 9 Combine like terms: -9 + 9 = 0 0 + x = -2.549509757 + 9 x = -2.549509757 + 9 Combine like terms: -2.549509757 + 9 = 6.450490243 x = 6.450490243 Simplifying x = 6.450490243Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.549509757, 6.450490243}Solution
x = {0, 11.549509757, 6.450490243}
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