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Simplifying 1000 = 2x2 + 9x + 4 Reorder the terms: 1000 = 4 + 9x + 2x2 Solving 1000 = 4 + 9x + 2x2 Solving for variable 'x'. Combine like terms: 1000 + -4 = 996 996 + -9x + -2x2 = 4 + 9x + 2x2 + -4 + -9x + -2x2 Reorder the terms: 996 + -9x + -2x2 = 4 + -4 + 9x + -9x + 2x2 + -2x2 Combine like terms: 4 + -4 = 0 996 + -9x + -2x2 = 0 + 9x + -9x + 2x2 + -2x2 996 + -9x + -2x2 = 9x + -9x + 2x2 + -2x2 Combine like terms: 9x + -9x = 0 996 + -9x + -2x2 = 0 + 2x2 + -2x2 996 + -9x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 996 + -9x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -498 + 4.5x + x2 = 0 Move the constant term to the right: Add '498' to each side of the equation. -498 + 4.5x + 498 + x2 = 0 + 498 Reorder the terms: -498 + 498 + 4.5x + x2 = 0 + 498 Combine like terms: -498 + 498 = 0 0 + 4.5x + x2 = 0 + 498 4.5x + x2 = 0 + 498 Combine like terms: 0 + 498 = 498 4.5x + x2 = 498 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 498 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 498 + 5.0625 Combine like terms: 498 + 5.0625 = 503.0625 5.0625 + 4.5x + x2 = 503.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 503.0625 Calculate the square root of the right side: 22.429054817 Break this problem into two subproblems by setting (x + 2.25) equal to 22.429054817 and -22.429054817.Subproblem 1
x + 2.25 = 22.429054817 Simplifying x + 2.25 = 22.429054817 Reorder the terms: 2.25 + x = 22.429054817 Solving 2.25 + x = 22.429054817 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 22.429054817 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 22.429054817 + -2.25 x = 22.429054817 + -2.25 Combine like terms: 22.429054817 + -2.25 = 20.179054817 x = 20.179054817 Simplifying x = 20.179054817Subproblem 2
x + 2.25 = -22.429054817 Simplifying x + 2.25 = -22.429054817 Reorder the terms: 2.25 + x = -22.429054817 Solving 2.25 + x = -22.429054817 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -22.429054817 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -22.429054817 + -2.25 x = -22.429054817 + -2.25 Combine like terms: -22.429054817 + -2.25 = -24.679054817 x = -24.679054817 Simplifying x = -24.679054817Solution
The solution to the problem is based on the solutions from the subproblems. x = {20.179054817, -24.679054817}
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