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Simplifying 250 = (3x + -1)(2x + 5) Reorder the terms: 250 = (-1 + 3x)(2x + 5) Reorder the terms: 250 = (-1 + 3x)(5 + 2x) Multiply (-1 + 3x) * (5 + 2x) 250 = (-1(5 + 2x) + 3x * (5 + 2x)) 250 = ((5 * -1 + 2x * -1) + 3x * (5 + 2x)) 250 = ((-5 + -2x) + 3x * (5 + 2x)) 250 = (-5 + -2x + (5 * 3x + 2x * 3x)) 250 = (-5 + -2x + (15x + 6x2)) Combine like terms: -2x + 15x = 13x 250 = (-5 + 13x + 6x2) Solving 250 = -5 + 13x + 6x2 Solving for variable 'x'. Combine like terms: 250 + 5 = 255 255 + -13x + -6x2 = -5 + 13x + 6x2 + 5 + -13x + -6x2 Reorder the terms: 255 + -13x + -6x2 = -5 + 5 + 13x + -13x + 6x2 + -6x2 Combine like terms: -5 + 5 = 0 255 + -13x + -6x2 = 0 + 13x + -13x + 6x2 + -6x2 255 + -13x + -6x2 = 13x + -13x + 6x2 + -6x2 Combine like terms: 13x + -13x = 0 255 + -13x + -6x2 = 0 + 6x2 + -6x2 255 + -13x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 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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