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Simplifying x4 + -52500x2 + 1250000 = 0 Reorder the terms: 1250000 + -52500x2 + x4 = 0 Solving 1250000 + -52500x2 + x4 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-1250000' to each side of the equation. 1250000 + -52500x2 + -1250000 + x4 = 0 + -1250000 Reorder the terms: 1250000 + -1250000 + -52500x2 + x4 = 0 + -1250000 Combine like terms: 1250000 + -1250000 = 0 0 + -52500x2 + x4 = 0 + -1250000 -52500x2 + x4 = 0 + -1250000 Combine like terms: 0 + -1250000 = -1250000 -52500x2 + x4 = -1250000 The x term is -52500x2. Take half its coefficient (-26250). Square it (689062500) and add it to both sides. Add '689062500' to each side of the equation. -52500x2 + 689062500 + x4 = -1250000 + 689062500 Reorder the terms: 689062500 + -52500x2 + x4 = -1250000 + 689062500 Combine like terms: -1250000 + 689062500 = 687812500 689062500 + -52500x2 + x4 = 687812500 Factor a perfect square on the left side: (x2 + -26250)(x2 + -26250) = 687812500 Calculate the square root of the right side: 26226.179668415 Break this problem into two subproblems by setting (x2 + -26250) equal to 26226.179668415 and -26226.179668415.Subproblem 1
x2 + -26250 = 26226.179668415 Simplifying x2 + -26250 = 26226.179668415 Reorder the terms: -26250 + x2 = 26226.179668415 Solving -26250 + x2 = 26226.179668415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '26250' to each side of the equation. -26250 + 26250 + x2 = 26226.179668415 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = 26226.179668415 + 26250 x2 = 26226.179668415 + 26250 Combine like terms: 26226.179668415 + 26250 = 52476.179668415 x2 = 52476.179668415 Simplifying x2 = 52476.179668415 Take the square root of each side: x = {-229.076798625, 229.076798625}Subproblem 2
x2 + -26250 = -26226.179668415 Simplifying x2 + -26250 = -26226.179668415 Reorder the terms: -26250 + x2 = -26226.179668415 Solving -26250 + x2 = -26226.179668415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '26250' to each side of the equation. -26250 + 26250 + x2 = -26226.179668415 + 26250 Combine like terms: -26250 + 26250 = 0 0 + x2 = -26226.179668415 + 26250 x2 = -26226.179668415 + 26250 Combine like terms: -26226.179668415 + 26250 = 23.820331585 x2 = 23.820331585 Simplifying x2 = 23.820331585 Take the square root of each side: x = {-4.880607707, 4.880607707}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-229.076798625, 229.076798625, -4.880607707, 4.880607707}
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