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Simplifying 0 = 0.5X2 + -55X + 1000 Reorder the terms: 0 = 1000 + -55X + 0.5X2 Solving 0 = 1000 + -55X + 0.5X2 Solving for variable 'X'. Combine like terms: 0 + -1000 = -1000 -1000 + 55X + -0.5X2 = 1000 + -55X + 0.5X2 + -1000 + 55X + -0.5X2 Reorder the terms: -1000 + 55X + -0.5X2 = 1000 + -1000 + -55X + 55X + 0.5X2 + -0.5X2 Combine like terms: 1000 + -1000 = 0 -1000 + 55X + -0.5X2 = 0 + -55X + 55X + 0.5X2 + -0.5X2 -1000 + 55X + -0.5X2 = -55X + 55X + 0.5X2 + -0.5X2 Combine like terms: -55X + 55X = 0 -1000 + 55X + -0.5X2 = 0 + 0.5X2 + -0.5X2 -1000 + 55X + -0.5X2 = 0.5X2 + -0.5X2 Combine like terms: 0.5X2 + -0.5X2 = 0.0 -1000 + 55X + -0.5X2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 2000 + -110X + X2 = 0 Move the constant term to the right: Add '-2000' to each side of the equation. 2000 + -110X + -2000 + X2 = 0 + -2000 Reorder the terms: 2000 + -2000 + -110X + X2 = 0 + -2000 Combine like terms: 2000 + -2000 = 0 0 + -110X + X2 = 0 + -2000 -110X + X2 = 0 + -2000 Combine like terms: 0 + -2000 = -2000 -110X + X2 = -2000 The X term is -110X. Take half its coefficient (-55). Square it (3025) and add it to both sides. Add '3025' to each side of the equation. -110X + 3025 + X2 = -2000 + 3025 Reorder the terms: 3025 + -110X + X2 = -2000 + 3025 Combine like terms: -2000 + 3025 = 1025 3025 + -110X + X2 = 1025 Factor a perfect square on the left side: (X + -55)(X + -55) = 1025 Calculate the square root of the right side: 32.015621187 Break this problem into two subproblems by setting (X + -55) equal to 32.015621187 and -32.015621187.Subproblem 1
X + -55 = 32.015621187 Simplifying X + -55 = 32.015621187 Reorder the terms: -55 + X = 32.015621187 Solving -55 + X = 32.015621187 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '55' to each side of the equation. -55 + 55 + X = 32.015621187 + 55 Combine like terms: -55 + 55 = 0 0 + X = 32.015621187 + 55 X = 32.015621187 + 55 Combine like terms: 32.015621187 + 55 = 87.015621187 X = 87.015621187 Simplifying X = 87.015621187Subproblem 2
X + -55 = -32.015621187 Simplifying X + -55 = -32.015621187 Reorder the terms: -55 + X = -32.015621187 Solving -55 + X = -32.015621187 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '55' to each side of the equation. -55 + 55 + X = -32.015621187 + 55 Combine like terms: -55 + 55 = 0 0 + X = -32.015621187 + 55 X = -32.015621187 + 55 Combine like terms: -32.015621187 + 55 = 22.984378813 X = 22.984378813 Simplifying X = 22.984378813Solution
The solution to the problem is based on the solutions from the subproblems. X = {87.015621187, 22.984378813}
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