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Simplifying 500000 = -2x2 + 4000x Reorder the terms: 500000 = 4000x + -2x2 Solving 500000 = 4000x + -2x2 Solving for variable 'x'. Reorder the terms: 500000 + -4000x + 2x2 = 4000x + -4000x + -2x2 + 2x2 Combine like terms: 4000x + -4000x = 0 500000 + -4000x + 2x2 = 0 + -2x2 + 2x2 500000 + -4000x + 2x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 500000 + -4000x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(250000 + -2000x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(250000 + -2000x + x2)' equal to zero and attempt to solve: Simplifying 250000 + -2000x + x2 = 0 Solving 250000 + -2000x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-250000' to each side of the equation. 250000 + -2000x + -250000 + x2 = 0 + -250000 Reorder the terms: 250000 + -250000 + -2000x + x2 = 0 + -250000 Combine like terms: 250000 + -250000 = 0 0 + -2000x + x2 = 0 + -250000 -2000x + x2 = 0 + -250000 Combine like terms: 0 + -250000 = -250000 -2000x + x2 = -250000 The x term is -2000x. Take half its coefficient (-1000). Square it (1000000) and add it to both sides. Add '1000000' to each side of the equation. -2000x + 1000000 + x2 = -250000 + 1000000 Reorder the terms: 1000000 + -2000x + x2 = -250000 + 1000000 Combine like terms: -250000 + 1000000 = 750000 1000000 + -2000x + x2 = 750000 Factor a perfect square on the left side: (x + -1000)(x + -1000) = 750000 Calculate the square root of the right side: 866.025403784 Break this problem into two subproblems by setting (x + -1000) equal to 866.025403784 and -866.025403784.Subproblem 1
x + -1000 = 866.025403784 Simplifying x + -1000 = 866.025403784 Reorder the terms: -1000 + x = 866.025403784 Solving -1000 + x = 866.025403784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + x = 866.025403784 + 1000 Combine like terms: -1000 + 1000 = 0 0 + x = 866.025403784 + 1000 x = 866.025403784 + 1000 Combine like terms: 866.025403784 + 1000 = 1866.025403784 x = 1866.025403784 Simplifying x = 1866.025403784Subproblem 2
x + -1000 = -866.025403784 Simplifying x + -1000 = -866.025403784 Reorder the terms: -1000 + x = -866.025403784 Solving -1000 + x = -866.025403784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + x = -866.025403784 + 1000 Combine like terms: -1000 + 1000 = 0 0 + x = -866.025403784 + 1000 x = -866.025403784 + 1000 Combine like terms: -866.025403784 + 1000 = 133.974596216 x = 133.974596216 Simplifying x = 133.974596216Solution
The solution to the problem is based on the solutions from the subproblems. x = {1866.025403784, 133.974596216}Solution
x = {1866.025403784, 133.974596216}
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