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Simplifying 11000000 = 0.5x2 + 10x + 7000 Reorder the terms: 11000000 = 7000 + 10x + 0.5x2 Solving 11000000 = 7000 + 10x + 0.5x2 Solving for variable 'x'. Combine like terms: 11000000 + -7000 = 10993000 10993000 + -10x + -0.5x2 = 7000 + 10x + 0.5x2 + -7000 + -10x + -0.5x2 Reorder the terms: 10993000 + -10x + -0.5x2 = 7000 + -7000 + 10x + -10x + 0.5x2 + -0.5x2 Combine like terms: 7000 + -7000 = 0 10993000 + -10x + -0.5x2 = 0 + 10x + -10x + 0.5x2 + -0.5x2 10993000 + -10x + -0.5x2 = 10x + -10x + 0.5x2 + -0.5x2 Combine like terms: 10x + -10x = 0 10993000 + -10x + -0.5x2 = 0 + 0.5x2 + -0.5x2 10993000 + -10x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 10993000 + -10x + -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'. -21986000 + 20x + x2 = 0 Move the constant term to the right: Add '21986000' to each side of the equation. -21986000 + 20x + 21986000 + x2 = 0 + 21986000 Reorder the terms: -21986000 + 21986000 + 20x + x2 = 0 + 21986000 Combine like terms: -21986000 + 21986000 = 0 0 + 20x + x2 = 0 + 21986000 20x + x2 = 0 + 21986000 Combine like terms: 0 + 21986000 = 21986000 20x + x2 = 21986000 The x term is 20x. Take half its coefficient (10). Square it (100) and add it to both sides. Add '100' to each side of the equation. 20x + 100 + x2 = 21986000 + 100 Reorder the terms: 100 + 20x + x2 = 21986000 + 100 Combine like terms: 21986000 + 100 = 21986100 100 + 20x + x2 = 21986100 Factor a perfect square on the left side: (x + 10)(x + 10) = 21986100 Calculate the square root of the right side: 4688.933780722 Break this problem into two subproblems by setting (x + 10) equal to 4688.933780722 and -4688.933780722.Subproblem 1
x + 10 = 4688.933780722 Simplifying x + 10 = 4688.933780722 Reorder the terms: 10 + x = 4688.933780722 Solving 10 + x = 4688.933780722 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = 4688.933780722 + -10 Combine like terms: 10 + -10 = 0 0 + x = 4688.933780722 + -10 x = 4688.933780722 + -10 Combine like terms: 4688.933780722 + -10 = 4678.933780722 x = 4678.933780722 Simplifying x = 4678.933780722Subproblem 2
x + 10 = -4688.933780722 Simplifying x + 10 = -4688.933780722 Reorder the terms: 10 + x = -4688.933780722 Solving 10 + x = -4688.933780722 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = -4688.933780722 + -10 Combine like terms: 10 + -10 = 0 0 + x = -4688.933780722 + -10 x = -4688.933780722 + -10 Combine like terms: -4688.933780722 + -10 = -4698.933780722 x = -4698.933780722 Simplifying x = -4698.933780722Solution
The solution to the problem is based on the solutions from the subproblems. x = {4678.933780722, -4698.933780722}
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