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Simplifying x + 3x(x + 5000) = 40000 Reorder the terms: x + 3x(5000 + x) = 40000 x + (5000 * 3x + x * 3x) = 40000 x + (15000x + 3x2) = 40000 Combine like terms: x + 15000x = 15001x 15001x + 3x2 = 40000 Solving 15001x + 3x2 = 40000 Solving for variable 'x'. Reorder the terms: -40000 + 15001x + 3x2 = 40000 + -40000 Combine like terms: 40000 + -40000 = 0 -40000 + 15001x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -13333.33333 + 5000.333333x + x2 = 0 Move the constant term to the right: Add '13333.33333' to each side of the equation. -13333.33333 + 5000.333333x + 13333.33333 + x2 = 0 + 13333.33333 Reorder the terms: -13333.33333 + 13333.33333 + 5000.333333x + x2 = 0 + 13333.33333 Combine like terms: -13333.33333 + 13333.33333 = 0.00000 0.00000 + 5000.333333x + x2 = 0 + 13333.33333 5000.333333x + x2 = 0 + 13333.33333 Combine like terms: 0 + 13333.33333 = 13333.33333 5000.333333x + x2 = 13333.33333 The x term is 5000.333333x. Take half its coefficient (2500.166667). Square it (6250833.363) and add it to both sides. Add '6250833.363' to each side of the equation. 5000.333333x + 6250833.363 + x2 = 13333.33333 + 6250833.363 Reorder the terms: 6250833.363 + 5000.333333x + x2 = 13333.33333 + 6250833.363 Combine like terms: 13333.33333 + 6250833.363 = 6264166.69633 6250833.363 + 5000.333333x + x2 = 6264166.69633 Factor a perfect square on the left side: (x + 2500.166667)(x + 2500.166667) = 6264166.69633 Calculate the square root of the right side: 2502.831735521 Break this problem into two subproblems by setting (x + 2500.166667) equal to 2502.831735521 and -2502.831735521.Subproblem 1
x + 2500.166667 = 2502.831735521 Simplifying x + 2500.166667 = 2502.831735521 Reorder the terms: 2500.166667 + x = 2502.831735521 Solving 2500.166667 + x = 2502.831735521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2500.166667' to each side of the equation. 2500.166667 + -2500.166667 + x = 2502.831735521 + -2500.166667 Combine like terms: 2500.166667 + -2500.166667 = 0.000000 0.000000 + x = 2502.831735521 + -2500.166667 x = 2502.831735521 + -2500.166667 Combine like terms: 2502.831735521 + -2500.166667 = 2.665068521 x = 2.665068521 Simplifying x = 2.665068521Subproblem 2
x + 2500.166667 = -2502.831735521 Simplifying x + 2500.166667 = -2502.831735521 Reorder the terms: 2500.166667 + x = -2502.831735521 Solving 2500.166667 + x = -2502.831735521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2500.166667' to each side of the equation. 2500.166667 + -2500.166667 + x = -2502.831735521 + -2500.166667 Combine like terms: 2500.166667 + -2500.166667 = 0.000000 0.000000 + x = -2502.831735521 + -2500.166667 x = -2502.831735521 + -2500.166667 Combine like terms: -2502.831735521 + -2500.166667 = -5002.998402521 x = -5002.998402521 Simplifying x = -5002.998402521Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.665068521, -5002.998402521}
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