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Simplifying 0 = -32x2 + 2025x + 243000 Reorder the terms: 0 = 243000 + 2025x + -32x2 Solving 0 = 243000 + 2025x + -32x2 Solving for variable 'x'. Combine like terms: 0 + -243000 = -243000 -243000 + -2025x + 32x2 = 243000 + 2025x + -32x2 + -243000 + -2025x + 32x2 Reorder the terms: -243000 + -2025x + 32x2 = 243000 + -243000 + 2025x + -2025x + -32x2 + 32x2 Combine like terms: 243000 + -243000 = 0 -243000 + -2025x + 32x2 = 0 + 2025x + -2025x + -32x2 + 32x2 -243000 + -2025x + 32x2 = 2025x + -2025x + -32x2 + 32x2 Combine like terms: 2025x + -2025x = 0 -243000 + -2025x + 32x2 = 0 + -32x2 + 32x2 -243000 + -2025x + 32x2 = -32x2 + 32x2 Combine like terms: -32x2 + 32x2 = 0 -243000 + -2025x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -7593.75 + -63.28125x + x2 = 0 Move the constant term to the right: Add '7593.75' to each side of the equation. -7593.75 + -63.28125x + 7593.75 + x2 = 0 + 7593.75 Reorder the terms: -7593.75 + 7593.75 + -63.28125x + x2 = 0 + 7593.75 Combine like terms: -7593.75 + 7593.75 = 0.00 0.00 + -63.28125x + x2 = 0 + 7593.75 -63.28125x + x2 = 0 + 7593.75 Combine like terms: 0 + 7593.75 = 7593.75 -63.28125x + x2 = 7593.75 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 = 7593.75 + 1001.129150 Reorder the terms: 1001.129150 + -63.28125x + x2 = 7593.75 + 1001.129150 Combine like terms: 7593.75 + 1001.129150 = 8594.87915 1001.129150 + -63.28125x + x2 = 8594.87915 Factor a perfect square on the left side: (x + -31.640625)(x + -31.640625) = 8594.87915 Calculate the square root of the right side: 92.708571071 Break this problem into two subproblems by setting (x + -31.640625) equal to 92.708571071 and -92.708571071.Subproblem 1
x + -31.640625 = 92.708571071 Simplifying x + -31.640625 = 92.708571071 Reorder the terms: -31.640625 + x = 92.708571071 Solving -31.640625 + x = 92.708571071 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 = 92.708571071 + 31.640625 Combine like terms: -31.640625 + 31.640625 = 0.000000 0.000000 + x = 92.708571071 + 31.640625 x = 92.708571071 + 31.640625 Combine like terms: 92.708571071 + 31.640625 = 124.349196071 x = 124.349196071 Simplifying x = 124.349196071Subproblem 2
x + -31.640625 = -92.708571071 Simplifying x + -31.640625 = -92.708571071 Reorder the terms: -31.640625 + x = -92.708571071 Solving -31.640625 + x = -92.708571071 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 = -92.708571071 + 31.640625 Combine like terms: -31.640625 + 31.640625 = 0.000000 0.000000 + x = -92.708571071 + 31.640625 x = -92.708571071 + 31.640625 Combine like terms: -92.708571071 + 31.640625 = -61.067946071 x = -61.067946071 Simplifying x = -61.067946071Solution
The solution to the problem is based on the solutions from the subproblems. x = {124.349196071, -61.067946071}
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