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Simplifying 15000 = 25X(X + -75) + 50X Reorder the terms: 15000 = 25X(-75 + X) + 50X 15000 = (-75 * 25X + X * 25X) + 50X 15000 = (-1875X + 25X2) + 50X Reorder the terms: 15000 = -1875X + 50X + 25X2 Combine like terms: -1875X + 50X = -1825X 15000 = -1825X + 25X2 Solving 15000 = -1825X + 25X2 Solving for variable 'X'. Reorder the terms: 15000 + 1825X + -25X2 = -1825X + 1825X + 25X2 + -25X2 Combine like terms: -1825X + 1825X = 0 15000 + 1825X + -25X2 = 0 + 25X2 + -25X2 15000 + 1825X + -25X2 = 25X2 + -25X2 Combine like terms: 25X2 + -25X2 = 0 15000 + 1825X + -25X2 = 0 Factor out the Greatest Common Factor (GCF), '25'. 25(600 + 73X + -1X2) = 0 Ignore the factor 25.Subproblem 1
Set the factor '(600 + 73X + -1X2)' equal to zero and attempt to solve: Simplifying 600 + 73X + -1X2 = 0 Solving 600 + 73X + -1X2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -600 + -73X + X2 = 0 Move the constant term to the right: Add '600' to each side of the equation. -600 + -73X + 600 + X2 = 0 + 600 Reorder the terms: -600 + 600 + -73X + X2 = 0 + 600 Combine like terms: -600 + 600 = 0 0 + -73X + X2 = 0 + 600 -73X + X2 = 0 + 600 Combine like terms: 0 + 600 = 600 -73X + X2 = 600 The X term is -73X. Take half its coefficient (-36.5). Square it (1332.25) and add it to both sides. Add '1332.25' to each side of the equation. -73X + 1332.25 + X2 = 600 + 1332.25 Reorder the terms: 1332.25 + -73X + X2 = 600 + 1332.25 Combine like terms: 600 + 1332.25 = 1932.25 1332.25 + -73X + X2 = 1932.25 Factor a perfect square on the left side: (X + -36.5)(X + -36.5) = 1932.25 Calculate the square root of the right side: 43.957365708 Break this problem into two subproblems by setting (X + -36.5) equal to 43.957365708 and -43.957365708.Subproblem 1
X + -36.5 = 43.957365708 Simplifying X + -36.5 = 43.957365708 Reorder the terms: -36.5 + X = 43.957365708 Solving -36.5 + X = 43.957365708 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '36.5' to each side of the equation. -36.5 + 36.5 + X = 43.957365708 + 36.5 Combine like terms: -36.5 + 36.5 = 0.0 0.0 + X = 43.957365708 + 36.5 X = 43.957365708 + 36.5 Combine like terms: 43.957365708 + 36.5 = 80.457365708 X = 80.457365708 Simplifying X = 80.457365708Subproblem 2
X + -36.5 = -43.957365708 Simplifying X + -36.5 = -43.957365708 Reorder the terms: -36.5 + X = -43.957365708 Solving -36.5 + X = -43.957365708 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '36.5' to each side of the equation. -36.5 + 36.5 + X = -43.957365708 + 36.5 Combine like terms: -36.5 + 36.5 = 0.0 0.0 + X = -43.957365708 + 36.5 X = -43.957365708 + 36.5 Combine like terms: -43.957365708 + 36.5 = -7.457365708 X = -7.457365708 Simplifying X = -7.457365708Solution
The solution to the problem is based on the solutions from the subproblems. X = {80.457365708, -7.457365708}Solution
X = {80.457365708, -7.457365708}
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