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Simplifying 500 = X(X + 1) Reorder the terms: 500 = X(1 + X) 500 = (1 * X + X * X) 500 = (1X + X2) Solving 500 = 1X + X2 Solving for variable 'X'. Reorder the terms: 500 + -1X + -1X2 = 1X + -1X + X2 + -1X2 Combine like terms: 1X + -1X = 0 500 + -1X + -1X2 = 0 + X2 + -1X2 500 + -1X + -1X2 = X2 + -1X2 Combine like terms: X2 + -1X2 = 0 500 + -1X + -1X2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -500 + X + X2 = 0 Move the constant term to the right: Add '500' to each side of the equation. -500 + X + 500 + X2 = 0 + 500 Reorder the terms: -500 + 500 + X + X2 = 0 + 500 Combine like terms: -500 + 500 = 0 0 + X + X2 = 0 + 500 X + X2 = 0 + 500 Combine like terms: 0 + 500 = 500 X + X2 = 500 The X term is X. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + X2 = 500 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + X2 = 500 + 0.25 Combine like terms: 500 + 0.25 = 500.25 1.25 + X2 = 500.25 Factor a perfect square on the left side: (X + 0.5)(X + 0.5) = 500.25 Calculate the square root of the right side: 22.366269246 Break this problem into two subproblems by setting (X + 0.5) equal to 22.366269246 and -22.366269246.Subproblem 1
X + 0.5 = 22.366269246 Simplifying X + 0.5 = 22.366269246 Reorder the terms: 0.5 + X = 22.366269246 Solving 0.5 + X = 22.366269246 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = 22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 22.366269246 + -0.5 X = 22.366269246 + -0.5 Combine like terms: 22.366269246 + -0.5 = 21.866269246 X = 21.866269246 Simplifying X = 21.866269246Subproblem 2
X + 0.5 = -22.366269246 Simplifying X + 0.5 = -22.366269246 Reorder the terms: 0.5 + X = -22.366269246 Solving 0.5 + X = -22.366269246 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = -22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -22.366269246 + -0.5 X = -22.366269246 + -0.5 Combine like terms: -22.366269246 + -0.5 = -22.866269246 X = -22.866269246 Simplifying X = -22.866269246Solution
The solution to the problem is based on the solutions from the subproblems. X = {21.866269246, -22.866269246}
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