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Simplifying (2x + 5)(3x + -1) = 250 Reorder the terms: (5 + 2x)(3x + -1) = 250 Reorder the terms: (5 + 2x)(-1 + 3x) = 250 Multiply (5 + 2x) * (-1 + 3x) (5(-1 + 3x) + 2x * (-1 + 3x)) = 250 ((-1 * 5 + 3x * 5) + 2x * (-1 + 3x)) = 250 ((-5 + 15x) + 2x * (-1 + 3x)) = 250 (-5 + 15x + (-1 * 2x + 3x * 2x)) = 250 (-5 + 15x + (-2x + 6x2)) = 250 Combine like terms: 15x + -2x = 13x (-5 + 13x + 6x2) = 250 Solving -5 + 13x + 6x2 = 250 Solving for variable 'x'. Reorder the terms: -5 + -250 + 13x + 6x2 = 250 + -250 Combine like terms: -5 + -250 = -255 -255 + 13x + 6x2 = 250 + -250 Combine like terms: 250 + -250 = 0 -255 + 13x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -42.5 + 2.166666667x + x2 = 0 Move the constant term to the right: Add '42.5' to each side of the equation. -42.5 + 2.166666667x + 42.5 + x2 = 0 + 42.5 Reorder the terms: -42.5 + 42.5 + 2.166666667x + x2 = 0 + 42.5 Combine like terms: -42.5 + 42.5 = 0.0 0.0 + 2.166666667x + x2 = 0 + 42.5 2.166666667x + x2 = 0 + 42.5 Combine like terms: 0 + 42.5 = 42.5 2.166666667x + x2 = 42.5 The x term is 2.166666667x. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667x + 1.173611113 + x2 = 42.5 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667x + x2 = 42.5 + 1.173611113 Combine like terms: 42.5 + 1.173611113 = 43.673611113 1.173611113 + 2.166666667x + x2 = 43.673611113 Factor a perfect square on the left side: (x + 1.083333334)(x + 1.083333334) = 43.673611113 Calculate the square root of the right side: 6.608601298 Break this problem into two subproblems by setting (x + 1.083333334) equal to 6.608601298 and -6.608601298.Subproblem 1
x + 1.083333334 = 6.608601298 Simplifying x + 1.083333334 = 6.608601298 Reorder the terms: 1.083333334 + x = 6.608601298 Solving 1.083333334 + x = 6.608601298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = 6.608601298 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = 6.608601298 + -1.083333334 x = 6.608601298 + -1.083333334 Combine like terms: 6.608601298 + -1.083333334 = 5.525267964 x = 5.525267964 Simplifying x = 5.525267964Subproblem 2
x + 1.083333334 = -6.608601298 Simplifying x + 1.083333334 = -6.608601298 Reorder the terms: 1.083333334 + x = -6.608601298 Solving 1.083333334 + x = -6.608601298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = -6.608601298 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = -6.608601298 + -1.083333334 x = -6.608601298 + -1.083333334 Combine like terms: -6.608601298 + -1.083333334 = -7.691934632 x = -7.691934632 Simplifying x = -7.691934632Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.525267964, -7.691934632}
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