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Simplifying (3x + -5)(2x + -1) = 40 Reorder the terms: (-5 + 3x)(2x + -1) = 40 Reorder the terms: (-5 + 3x)(-1 + 2x) = 40 Multiply (-5 + 3x) * (-1 + 2x) (-5(-1 + 2x) + 3x * (-1 + 2x)) = 40 ((-1 * -5 + 2x * -5) + 3x * (-1 + 2x)) = 40 ((5 + -10x) + 3x * (-1 + 2x)) = 40 (5 + -10x + (-1 * 3x + 2x * 3x)) = 40 (5 + -10x + (-3x + 6x2)) = 40 Combine like terms: -10x + -3x = -13x (5 + -13x + 6x2) = 40 Solving 5 + -13x + 6x2 = 40 Solving for variable 'x'. Reorder the terms: 5 + -40 + -13x + 6x2 = 40 + -40 Combine like terms: 5 + -40 = -35 -35 + -13x + 6x2 = 40 + -40 Combine like terms: 40 + -40 = 0 -35 + -13x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.833333333 + -2.166666667x + x2 = 0 Move the constant term to the right: Add '5.833333333' to each side of the equation. -5.833333333 + -2.166666667x + 5.833333333 + x2 = 0 + 5.833333333 Reorder the terms: -5.833333333 + 5.833333333 + -2.166666667x + x2 = 0 + 5.833333333 Combine like terms: -5.833333333 + 5.833333333 = 0.000000000 0.000000000 + -2.166666667x + x2 = 0 + 5.833333333 -2.166666667x + x2 = 0 + 5.833333333 Combine like terms: 0 + 5.833333333 = 5.833333333 -2.166666667x + x2 = 5.833333333 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 = 5.833333333 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667x + x2 = 5.833333333 + 1.173611113 Combine like terms: 5.833333333 + 1.173611113 = 7.006944446 1.173611113 + -2.166666667x + x2 = 7.006944446 Factor a perfect square on the left side: (x + -1.083333334)(x + -1.083333334) = 7.006944446 Calculate the square root of the right side: 2.647063363 Break this problem into two subproblems by setting (x + -1.083333334) equal to 2.647063363 and -2.647063363.Subproblem 1
x + -1.083333334 = 2.647063363 Simplifying x + -1.083333334 = 2.647063363 Reorder the terms: -1.083333334 + x = 2.647063363 Solving -1.083333334 + x = 2.647063363 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 = 2.647063363 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = 2.647063363 + 1.083333334 x = 2.647063363 + 1.083333334 Combine like terms: 2.647063363 + 1.083333334 = 3.730396697 x = 3.730396697 Simplifying x = 3.730396697Subproblem 2
x + -1.083333334 = -2.647063363 Simplifying x + -1.083333334 = -2.647063363 Reorder the terms: -1.083333334 + x = -2.647063363 Solving -1.083333334 + x = -2.647063363 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 = -2.647063363 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = -2.647063363 + 1.083333334 x = -2.647063363 + 1.083333334 Combine like terms: -2.647063363 + 1.083333334 = -1.563730029 x = -1.563730029 Simplifying x = -1.563730029Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.730396697, -1.563730029}
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