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Simplifying 2x + -100 + x(2x + -100) * 1000 + x = 5000 Reorder the terms: 2x + -100 + x(-100 + 2x) * 1000 + x = 5000 Reorder the terms for easier multiplication: 2x + -100 + 1000x(-100 + 2x) + x = 5000 2x + -100 + (-100 * 1000x + 2x * 1000x) + x = 5000 2x + -100 + (-100000x + 2000x2) + x = 5000 Reorder the terms: -100 + 2x + -100000x + x + 2000x2 = 5000 Combine like terms: 2x + -100000x = -99998x -100 + -99998x + x + 2000x2 = 5000 Combine like terms: -99998x + x = -99997x -100 + -99997x + 2000x2 = 5000 Solving -100 + -99997x + 2000x2 = 5000 Solving for variable 'x'. Reorder the terms: -100 + -5000 + -99997x + 2000x2 = 5000 + -5000 Combine like terms: -100 + -5000 = -5100 -5100 + -99997x + 2000x2 = 5000 + -5000 Combine like terms: 5000 + -5000 = 0 -5100 + -99997x + 2000x2 = 0 Begin completing the square. Divide all terms by 2000 the coefficient of the squared term: Divide each side by '2000'. -2.55 + -49.9985x + x2 = 0.000 Move the constant term to the right: Add '2.55' to each side of the equation. -2.55 + -49.9985x + 2.55 + x2 = 0.000 + 2.55 Reorder the terms: -2.55 + 2.55 + -49.9985x + x2 = 0.000 + 2.55 Combine like terms: -2.55 + 2.55 = 0.00 0.00 + -49.9985x + x2 = 0.000 + 2.55 -49.9985x + x2 = 0.000 + 2.55 Combine like terms: 0.000 + 2.55 = 2.55 -49.9985x + x2 = 2.55 The x term is -49.9985x. Take half its coefficient (-24.99925). Square it (624.9625006) and add it to both sides. Add '624.9625006' to each side of the equation. -49.9985x + 624.9625006 + x2 = 2.55 + 624.9625006 Reorder the terms: 624.9625006 + -49.9985x + x2 = 2.55 + 624.9625006 Combine like terms: 2.55 + 624.9625006 = 627.5125006 624.9625006 + -49.9985x + x2 = 627.5125006 Factor a perfect square on the left side: (x + -24.99925)(x + -24.99925) = 627.5125006 Calculate the square root of the right side: 25.050199612 Break this problem into two subproblems by setting (x + -24.99925) equal to 25.050199612 and -25.050199612.Subproblem 1
x + -24.99925 = 25.050199612 Simplifying x + -24.99925 = 25.050199612 Reorder the terms: -24.99925 + x = 25.050199612 Solving -24.99925 + x = 25.050199612 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.99925' to each side of the equation. -24.99925 + 24.99925 + x = 25.050199612 + 24.99925 Combine like terms: -24.99925 + 24.99925 = 0.00000 0.00000 + x = 25.050199612 + 24.99925 x = 25.050199612 + 24.99925 Combine like terms: 25.050199612 + 24.99925 = 50.049449612 x = 50.049449612 Simplifying x = 50.049449612Subproblem 2
x + -24.99925 = -25.050199612 Simplifying x + -24.99925 = -25.050199612 Reorder the terms: -24.99925 + x = -25.050199612 Solving -24.99925 + x = -25.050199612 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.99925' to each side of the equation. -24.99925 + 24.99925 + x = -25.050199612 + 24.99925 Combine like terms: -24.99925 + 24.99925 = 0.00000 0.00000 + x = -25.050199612 + 24.99925 x = -25.050199612 + 24.99925 Combine like terms: -25.050199612 + 24.99925 = -0.050949612 x = -0.050949612 Simplifying x = -0.050949612Solution
The solution to the problem is based on the solutions from the subproblems. x = {50.049449612, -0.050949612}
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