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Simplifying -32x2 + 2500x + 450000 = 100 Reorder the terms: 450000 + 2500x + -32x2 = 100 Solving 450000 + 2500x + -32x2 = 100 Solving for variable 'x'. Reorder the terms: 450000 + -100 + 2500x + -32x2 = 100 + -100 Combine like terms: 450000 + -100 = 449900 449900 + 2500x + -32x2 = 100 + -100 Combine like terms: 100 + -100 = 0 449900 + 2500x + -32x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(112475 + 625x + -8x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(112475 + 625x + -8x2)' equal to zero and attempt to solve: Simplifying 112475 + 625x + -8x2 = 0 Solving 112475 + 625x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -14059.375 + -78.125x + x2 = 0 Move the constant term to the right: Add '14059.375' to each side of the equation. -14059.375 + -78.125x + 14059.375 + x2 = 0 + 14059.375 Reorder the terms: -14059.375 + 14059.375 + -78.125x + x2 = 0 + 14059.375 Combine like terms: -14059.375 + 14059.375 = 0.000 0.000 + -78.125x + x2 = 0 + 14059.375 -78.125x + x2 = 0 + 14059.375 Combine like terms: 0 + 14059.375 = 14059.375 -78.125x + x2 = 14059.375 The x term is -78.125x. Take half its coefficient (-39.0625). Square it (1525.878906) and add it to both sides. Add '1525.878906' to each side of the equation. -78.125x + 1525.878906 + x2 = 14059.375 + 1525.878906 Reorder the terms: 1525.878906 + -78.125x + x2 = 14059.375 + 1525.878906 Combine like terms: 14059.375 + 1525.878906 = 15585.253906 1525.878906 + -78.125x + x2 = 15585.253906 Factor a perfect square on the left side: (x + -39.0625)(x + -39.0625) = 15585.253906 Calculate the square root of the right side: 124.840914391 Break this problem into two subproblems by setting (x + -39.0625) equal to 124.840914391 and -124.840914391.Subproblem 1
x + -39.0625 = 124.840914391 Simplifying x + -39.0625 = 124.840914391 Reorder the terms: -39.0625 + x = 124.840914391 Solving -39.0625 + x = 124.840914391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '39.0625' to each side of the equation. -39.0625 + 39.0625 + x = 124.840914391 + 39.0625 Combine like terms: -39.0625 + 39.0625 = 0.0000 0.0000 + x = 124.840914391 + 39.0625 x = 124.840914391 + 39.0625 Combine like terms: 124.840914391 + 39.0625 = 163.903414391 x = 163.903414391 Simplifying x = 163.903414391Subproblem 2
x + -39.0625 = -124.840914391 Simplifying x + -39.0625 = -124.840914391 Reorder the terms: -39.0625 + x = -124.840914391 Solving -39.0625 + x = -124.840914391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '39.0625' to each side of the equation. -39.0625 + 39.0625 + x = -124.840914391 + 39.0625 Combine like terms: -39.0625 + 39.0625 = 0.0000 0.0000 + x = -124.840914391 + 39.0625 x = -124.840914391 + 39.0625 Combine like terms: -124.840914391 + 39.0625 = -85.778414391 x = -85.778414391 Simplifying x = -85.778414391Solution
The solution to the problem is based on the solutions from the subproblems. x = {163.903414391, -85.778414391}Solution
x = {163.903414391, -85.778414391}
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