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Simplifying 0 = -32x2 + 2025x + 162000 Reorder the terms: 0 = 162000 + 2025x + -32x2 Solving 0 = 162000 + 2025x + -32x2 Solving for variable 'x'. Combine like terms: 0 + -162000 = -162000 -162000 + -2025x + 32x2 = 162000 + 2025x + -32x2 + -162000 + -2025x + 32x2 Reorder the terms: -162000 + -2025x + 32x2 = 162000 + -162000 + 2025x + -2025x + -32x2 + 32x2 Combine like terms: 162000 + -162000 = 0 -162000 + -2025x + 32x2 = 0 + 2025x + -2025x + -32x2 + 32x2 -162000 + -2025x + 32x2 = 2025x + -2025x + -32x2 + 32x2 Combine like terms: 2025x + -2025x = 0 -162000 + -2025x + 32x2 = 0 + -32x2 + 32x2 -162000 + -2025x + 32x2 = -32x2 + 32x2 Combine like terms: -32x2 + 32x2 = 0 -162000 + -2025x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -5062.5 + -63.28125x + x2 = 0 Move the constant term to the right: Add '5062.5' to each side of the equation. -5062.5 + -63.28125x + 5062.5 + x2 = 0 + 5062.5 Reorder the terms: -5062.5 + 5062.5 + -63.28125x + x2 = 0 + 5062.5 Combine like terms: -5062.5 + 5062.5 = 0.0 0.0 + -63.28125x + x2 = 0 + 5062.5 -63.28125x + x2 = 0 + 5062.5 Combine like terms: 0 + 5062.5 = 5062.5 -63.28125x + x2 = 5062.5 The x term is -63.28125x. Take half its coefficient (-31.640625). Square it (1001.129150) and add it to both sides. Add '1001.129150' to each side of the equation. -63.28125x + 1001.129150 + x2 = 5062.5 + 1001.129150 Reorder the terms: 1001.129150 + -63.28125x + x2 = 5062.5 + 1001.129150 Combine like terms: 5062.5 + 1001.129150 = 6063.62915 1001.129150 + -63.28125x + x2 = 6063.62915 Factor a perfect square on the left side: (x + -31.640625)(x + -31.640625) = 6063.62915 Calculate the square root of the right side: 77.869308139 Break this problem into two subproblems by setting (x + -31.640625) equal to 77.869308139 and -77.869308139.Subproblem 1
x + -31.640625 = 77.869308139 Simplifying x + -31.640625 = 77.869308139 Reorder the terms: -31.640625 + x = 77.869308139 Solving -31.640625 + x = 77.869308139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.640625' to each side of the equation. -31.640625 + 31.640625 + x = 77.869308139 + 31.640625 Combine like terms: -31.640625 + 31.640625 = 0.000000 0.000000 + x = 77.869308139 + 31.640625 x = 77.869308139 + 31.640625 Combine like terms: 77.869308139 + 31.640625 = 109.509933139 x = 109.509933139 Simplifying x = 109.509933139Subproblem 2
x + -31.640625 = -77.869308139 Simplifying x + -31.640625 = -77.869308139 Reorder the terms: -31.640625 + x = -77.869308139 Solving -31.640625 + x = -77.869308139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.640625' to each side of the equation. -31.640625 + 31.640625 + x = -77.869308139 + 31.640625 Combine like terms: -31.640625 + 31.640625 = 0.000000 0.000000 + x = -77.869308139 + 31.640625 x = -77.869308139 + 31.640625 Combine like terms: -77.869308139 + 31.640625 = -46.228683139 x = -46.228683139 Simplifying x = -46.228683139Solution
The solution to the problem is based on the solutions from the subproblems. x = {109.509933139, -46.228683139}
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