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Simplifying (2x + -5)(3x + 1) = 3x Reorder the terms: (-5 + 2x)(3x + 1) = 3x Reorder the terms: (-5 + 2x)(1 + 3x) = 3x Multiply (-5 + 2x) * (1 + 3x) (-5(1 + 3x) + 2x * (1 + 3x)) = 3x ((1 * -5 + 3x * -5) + 2x * (1 + 3x)) = 3x ((-5 + -15x) + 2x * (1 + 3x)) = 3x (-5 + -15x + (1 * 2x + 3x * 2x)) = 3x (-5 + -15x + (2x + 6x2)) = 3x Combine like terms: -15x + 2x = -13x (-5 + -13x + 6x2) = 3x Solving -5 + -13x + 6x2 = 3x Solving for variable 'x'. Reorder the terms: -5 + -13x + -3x + 6x2 = 3x + -3x Combine like terms: -13x + -3x = -16x -5 + -16x + 6x2 = 3x + -3x Combine like terms: 3x + -3x = 0 -5 + -16x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -2.666666667x + 0.8333333333 + x2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -2.666666667x + x2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -2.666666667x + x2 = 0 + 0.8333333333 -2.666666667x + x2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -2.666666667x + x2 = 0.8333333333 The x term is -2.666666667x. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667x + 1.777777780 + x2 = 0.8333333333 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = 0.8333333333 + 1.777777780 Combine like terms: 0.8333333333 + 1.777777780 = 2.6111111133 1.777777780 + -2.666666667x + x2 = 2.6111111133 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 2.6111111133 Calculate the square root of the right side: 1.615893286 Break this problem into two subproblems by setting (x + -1.333333334) equal to 1.615893286 and -1.615893286.Subproblem 1
x + -1.333333334 = 1.615893286 Simplifying x + -1.333333334 = 1.615893286 Reorder the terms: -1.333333334 + x = 1.615893286 Solving -1.333333334 + x = 1.615893286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.615893286 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.615893286 + 1.333333334 x = 1.615893286 + 1.333333334 Combine like terms: 1.615893286 + 1.333333334 = 2.94922662 x = 2.94922662 Simplifying x = 2.94922662Subproblem 2
x + -1.333333334 = -1.615893286 Simplifying x + -1.333333334 = -1.615893286 Reorder the terms: -1.333333334 + x = -1.615893286 Solving -1.333333334 + x = -1.615893286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.615893286 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.615893286 + 1.333333334 x = -1.615893286 + 1.333333334 Combine like terms: -1.615893286 + 1.333333334 = -0.282559952 x = -0.282559952 Simplifying x = -0.282559952Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.94922662, -0.282559952}
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