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Simplifying 500x + -2x2 = 300x + 2000 Reorder the terms: 500x + -2x2 = 2000 + 300x Solving 500x + -2x2 = 2000 + 300x Solving for variable 'x'. Reorder the terms: -2000 + 500x + -300x + -2x2 = 2000 + 300x + -2000 + -300x Combine like terms: 500x + -300x = 200x -2000 + 200x + -2x2 = 2000 + 300x + -2000 + -300x Reorder the terms: -2000 + 200x + -2x2 = 2000 + -2000 + 300x + -300x Combine like terms: 2000 + -2000 = 0 -2000 + 200x + -2x2 = 0 + 300x + -300x -2000 + 200x + -2x2 = 300x + -300x Combine like terms: 300x + -300x = 0 -2000 + 200x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1000 + 100x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1000 + 100x + -1x2)' equal to zero and attempt to solve: Simplifying -1000 + 100x + -1x2 = 0 Solving -1000 + 100x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1000 + -100x + x2 = 0 Move the constant term to the right: Add '-1000' to each side of the equation. 1000 + -100x + -1000 + x2 = 0 + -1000 Reorder the terms: 1000 + -1000 + -100x + x2 = 0 + -1000 Combine like terms: 1000 + -1000 = 0 0 + -100x + x2 = 0 + -1000 -100x + x2 = 0 + -1000 Combine like terms: 0 + -1000 = -1000 -100x + x2 = -1000 The x term is -100x. Take half its coefficient (-50). Square it (2500) and add it to both sides. Add '2500' to each side of the equation. -100x + 2500 + x2 = -1000 + 2500 Reorder the terms: 2500 + -100x + x2 = -1000 + 2500 Combine like terms: -1000 + 2500 = 1500 2500 + -100x + x2 = 1500 Factor a perfect square on the left side: (x + -50)(x + -50) = 1500 Calculate the square root of the right side: 38.729833462 Break this problem into two subproblems by setting (x + -50) equal to 38.729833462 and -38.729833462.Subproblem 1
x + -50 = 38.729833462 Simplifying x + -50 = 38.729833462 Reorder the terms: -50 + x = 38.729833462 Solving -50 + x = 38.729833462 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '50' to each side of the equation. -50 + 50 + x = 38.729833462 + 50 Combine like terms: -50 + 50 = 0 0 + x = 38.729833462 + 50 x = 38.729833462 + 50 Combine like terms: 38.729833462 + 50 = 88.729833462 x = 88.729833462 Simplifying x = 88.729833462Subproblem 2
x + -50 = -38.729833462 Simplifying x + -50 = -38.729833462 Reorder the terms: -50 + x = -38.729833462 Solving -50 + x = -38.729833462 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '50' to each side of the equation. -50 + 50 + x = -38.729833462 + 50 Combine like terms: -50 + 50 = 0 0 + x = -38.729833462 + 50 x = -38.729833462 + 50 Combine like terms: -38.729833462 + 50 = 11.270166538 x = 11.270166538 Simplifying x = 11.270166538Solution
The solution to the problem is based on the solutions from the subproblems. x = {88.729833462, 11.270166538}Solution
x = {88.729833462, 11.270166538}
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