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Simplifying 1000x = (100 + -1x)(10 + -1x) Multiply (100 + -1x) * (10 + -1x) 1000x = (100(10 + -1x) + -1x * (10 + -1x)) 1000x = ((10 * 100 + -1x * 100) + -1x * (10 + -1x)) 1000x = ((1000 + -100x) + -1x * (10 + -1x)) 1000x = (1000 + -100x + (10 * -1x + -1x * -1x)) 1000x = (1000 + -100x + (-10x + 1x2)) Combine like terms: -100x + -10x = -110x 1000x = (1000 + -110x + 1x2) Solving 1000x = 1000 + -110x + 1x2 Solving for variable 'x'. Reorder the terms: -1000 + 1000x + 110x + -1x2 = 1000 + -110x + 1x2 + -1000 + 110x + -1x2 Combine like terms: 1000x + 110x = 1110x -1000 + 1110x + -1x2 = 1000 + -110x + 1x2 + -1000 + 110x + -1x2 Reorder the terms: -1000 + 1110x + -1x2 = 1000 + -1000 + -110x + 110x + 1x2 + -1x2 Combine like terms: 1000 + -1000 = 0 -1000 + 1110x + -1x2 = 0 + -110x + 110x + 1x2 + -1x2 -1000 + 1110x + -1x2 = -110x + 110x + 1x2 + -1x2 Combine like terms: -110x + 110x = 0 -1000 + 1110x + -1x2 = 0 + 1x2 + -1x2 -1000 + 1110x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 -1000 + 1110x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1000 + -1110x + x2 = 0 Move the constant term to the right: Add '-1000' to each side of the equation. 1000 + -1110x + -1000 + x2 = 0 + -1000 Reorder the terms: 1000 + -1000 + -1110x + x2 = 0 + -1000 Combine like terms: 1000 + -1000 = 0 0 + -1110x + x2 = 0 + -1000 -1110x + x2 = 0 + -1000 Combine like terms: 0 + -1000 = -1000 -1110x + x2 = -1000 The x term is -1110x. Take half its coefficient (-555). Square it (308025) and add it to both sides. Add '308025' to each side of the equation. -1110x + 308025 + x2 = -1000 + 308025 Reorder the terms: 308025 + -1110x + x2 = -1000 + 308025 Combine like terms: -1000 + 308025 = 307025 308025 + -1110x + x2 = 307025 Factor a perfect square on the left side: (x + -555)(x + -555) = 307025 Calculate the square root of the right side: 554.098366718 Break this problem into two subproblems by setting (x + -555) equal to 554.098366718 and -554.098366718.Subproblem 1
x + -555 = 554.098366718 Simplifying x + -555 = 554.098366718 Reorder the terms: -555 + x = 554.098366718 Solving -555 + x = 554.098366718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '555' to each side of the equation. -555 + 555 + x = 554.098366718 + 555 Combine like terms: -555 + 555 = 0 0 + x = 554.098366718 + 555 x = 554.098366718 + 555 Combine like terms: 554.098366718 + 555 = 1109.098366718 x = 1109.098366718 Simplifying x = 1109.098366718Subproblem 2
x + -555 = -554.098366718 Simplifying x + -555 = -554.098366718 Reorder the terms: -555 + x = -554.098366718 Solving -555 + x = -554.098366718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '555' to each side of the equation. -555 + 555 + x = -554.098366718 + 555 Combine like terms: -555 + 555 = 0 0 + x = -554.098366718 + 555 x = -554.098366718 + 555 Combine like terms: -554.098366718 + 555 = 0.901633282 x = 0.901633282 Simplifying x = 0.901633282Solution
The solution to the problem is based on the solutions from the subproblems. x = {1109.098366718, 0.901633282}
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