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Simplifying 1 = -0.5 * 32x2 + 1000x + 50 Multiply -0.5 * 32 1 = -16x2 + 1000x + 50 Reorder the terms: 1 = 50 + 1000x + -16x2 Solving 1 = 50 + 1000x + -16x2 Solving for variable 'x'. Combine like terms: 1 + -50 = -49 -49 + -1000x + 16x2 = 50 + 1000x + -16x2 + -50 + -1000x + 16x2 Reorder the terms: -49 + -1000x + 16x2 = 50 + -50 + 1000x + -1000x + -16x2 + 16x2 Combine like terms: 50 + -50 = 0 -49 + -1000x + 16x2 = 0 + 1000x + -1000x + -16x2 + 16x2 -49 + -1000x + 16x2 = 1000x + -1000x + -16x2 + 16x2 Combine like terms: 1000x + -1000x = 0 -49 + -1000x + 16x2 = 0 + -16x2 + 16x2 -49 + -1000x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -49 + -1000x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -3.0625 + -62.5x + x2 = 0 Move the constant term to the right: Add '3.0625' to each side of the equation. -3.0625 + -62.5x + 3.0625 + x2 = 0 + 3.0625 Reorder the terms: -3.0625 + 3.0625 + -62.5x + x2 = 0 + 3.0625 Combine like terms: -3.0625 + 3.0625 = 0.0000 0.0000 + -62.5x + x2 = 0 + 3.0625 -62.5x + x2 = 0 + 3.0625 Combine like terms: 0 + 3.0625 = 3.0625 -62.5x + x2 = 3.0625 The x term is -62.5x. Take half its coefficient (-31.25). Square it (976.5625) and add it to both sides. Add '976.5625' to each side of the equation. -62.5x + 976.5625 + x2 = 3.0625 + 976.5625 Reorder the terms: 976.5625 + -62.5x + x2 = 3.0625 + 976.5625 Combine like terms: 3.0625 + 976.5625 = 979.625 976.5625 + -62.5x + x2 = 979.625 Factor a perfect square on the left side: (x + -31.25)(x + -31.25) = 979.625 Calculate the square root of the right side: 31.298961644 Break this problem into two subproblems by setting (x + -31.25) equal to 31.298961644 and -31.298961644.Subproblem 1
x + -31.25 = 31.298961644 Simplifying x + -31.25 = 31.298961644 Reorder the terms: -31.25 + x = 31.298961644 Solving -31.25 + x = 31.298961644 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.25' to each side of the equation. -31.25 + 31.25 + x = 31.298961644 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + x = 31.298961644 + 31.25 x = 31.298961644 + 31.25 Combine like terms: 31.298961644 + 31.25 = 62.548961644 x = 62.548961644 Simplifying x = 62.548961644Subproblem 2
x + -31.25 = -31.298961644 Simplifying x + -31.25 = -31.298961644 Reorder the terms: -31.25 + x = -31.298961644 Solving -31.25 + x = -31.298961644 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.25' to each side of the equation. -31.25 + 31.25 + x = -31.298961644 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + x = -31.298961644 + 31.25 x = -31.298961644 + 31.25 Combine like terms: -31.298961644 + 31.25 = -0.048961644 x = -0.048961644 Simplifying x = -0.048961644Solution
The solution to the problem is based on the solutions from the subproblems. x = {62.548961644, -0.048961644}
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