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Simplifying (3x + -7)(2x + -5) + -1x(4x + -3) = 0 Reorder the terms: (-7 + 3x)(2x + -5) + -1x(4x + -3) = 0 Reorder the terms: (-7 + 3x)(-5 + 2x) + -1x(4x + -3) = 0 Multiply (-7 + 3x) * (-5 + 2x) (-7(-5 + 2x) + 3x * (-5 + 2x)) + -1x(4x + -3) = 0 ((-5 * -7 + 2x * -7) + 3x * (-5 + 2x)) + -1x(4x + -3) = 0 ((35 + -14x) + 3x * (-5 + 2x)) + -1x(4x + -3) = 0 (35 + -14x + (-5 * 3x + 2x * 3x)) + -1x(4x + -3) = 0 (35 + -14x + (-15x + 6x2)) + -1x(4x + -3) = 0 Combine like terms: -14x + -15x = -29x (35 + -29x + 6x2) + -1x(4x + -3) = 0 Reorder the terms: 35 + -29x + 6x2 + -1x(-3 + 4x) = 0 35 + -29x + 6x2 + (-3 * -1x + 4x * -1x) = 0 35 + -29x + 6x2 + (3x + -4x2) = 0 Reorder the terms: 35 + -29x + 3x + 6x2 + -4x2 = 0 Combine like terms: -29x + 3x = -26x 35 + -26x + 6x2 + -4x2 = 0 Combine like terms: 6x2 + -4x2 = 2x2 35 + -26x + 2x2 = 0 Solving 35 + -26x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 17.5 + -13x + x2 = 0 Move the constant term to the right: Add '-17.5' to each side of the equation. 17.5 + -13x + -17.5 + x2 = 0 + -17.5 Reorder the terms: 17.5 + -17.5 + -13x + x2 = 0 + -17.5 Combine like terms: 17.5 + -17.5 = 0.0 0.0 + -13x + x2 = 0 + -17.5 -13x + x2 = 0 + -17.5 Combine like terms: 0 + -17.5 = -17.5 -13x + x2 = -17.5 The x term is -13x. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13x + 42.25 + x2 = -17.5 + 42.25 Reorder the terms: 42.25 + -13x + x2 = -17.5 + 42.25 Combine like terms: -17.5 + 42.25 = 24.75 42.25 + -13x + x2 = 24.75 Factor a perfect square on the left side: (x + -6.5)(x + -6.5) = 24.75 Calculate the square root of the right side: 4.974937186 Break this problem into two subproblems by setting (x + -6.5) equal to 4.974937186 and -4.974937186.Subproblem 1
x + -6.5 = 4.974937186 Simplifying x + -6.5 = 4.974937186 Reorder the terms: -6.5 + x = 4.974937186 Solving -6.5 + x = 4.974937186 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = 4.974937186 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = 4.974937186 + 6.5 x = 4.974937186 + 6.5 Combine like terms: 4.974937186 + 6.5 = 11.474937186 x = 11.474937186 Simplifying x = 11.474937186Subproblem 2
x + -6.5 = -4.974937186 Simplifying x + -6.5 = -4.974937186 Reorder the terms: -6.5 + x = -4.974937186 Solving -6.5 + x = -4.974937186 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + x = -4.974937186 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + x = -4.974937186 + 6.5 x = -4.974937186 + 6.5 Combine like terms: -4.974937186 + 6.5 = 1.525062814 x = 1.525062814 Simplifying x = 1.525062814Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.474937186, 1.525062814}
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