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Simplifying 4x2 + 5x + 10 = 1000 Reorder the terms: 10 + 5x + 4x2 = 1000 Solving 10 + 5x + 4x2 = 1000 Solving for variable 'x'. Reorder the terms: 10 + -1000 + 5x + 4x2 = 1000 + -1000 Combine like terms: 10 + -1000 = -990 -990 + 5x + 4x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -990 + 5x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -247.5 + 1.25x + x2 = 0 Move the constant term to the right: Add '247.5' to each side of the equation. -247.5 + 1.25x + 247.5 + x2 = 0 + 247.5 Reorder the terms: -247.5 + 247.5 + 1.25x + x2 = 0 + 247.5 Combine like terms: -247.5 + 247.5 = 0.0 0.0 + 1.25x + x2 = 0 + 247.5 1.25x + x2 = 0 + 247.5 Combine like terms: 0 + 247.5 = 247.5 1.25x + x2 = 247.5 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 247.5 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 247.5 + 0.390625 Combine like terms: 247.5 + 0.390625 = 247.890625 0.390625 + 1.25x + x2 = 247.890625 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 247.890625 Calculate the square root of the right side: 15.744542705 Break this problem into two subproblems by setting (x + 0.625) equal to 15.744542705 and -15.744542705.Subproblem 1
x + 0.625 = 15.744542705 Simplifying x + 0.625 = 15.744542705 Reorder the terms: 0.625 + x = 15.744542705 Solving 0.625 + x = 15.744542705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 15.744542705 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 15.744542705 + -0.625 x = 15.744542705 + -0.625 Combine like terms: 15.744542705 + -0.625 = 15.119542705 x = 15.119542705 Simplifying x = 15.119542705Subproblem 2
x + 0.625 = -15.744542705 Simplifying x + 0.625 = -15.744542705 Reorder the terms: 0.625 + x = -15.744542705 Solving 0.625 + x = -15.744542705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -15.744542705 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -15.744542705 + -0.625 x = -15.744542705 + -0.625 Combine like terms: -15.744542705 + -0.625 = -16.369542705 x = -16.369542705 Simplifying x = -16.369542705Solution
The solution to the problem is based on the solutions from the subproblems. x = {15.119542705, -16.369542705}
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