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Simplifying (2x + -3)(4x + -5) + -2 = 0 Reorder the terms: (-3 + 2x)(4x + -5) + -2 = 0 Reorder the terms: (-3 + 2x)(-5 + 4x) + -2 = 0 Multiply (-3 + 2x) * (-5 + 4x) (-3(-5 + 4x) + 2x * (-5 + 4x)) + -2 = 0 ((-5 * -3 + 4x * -3) + 2x * (-5 + 4x)) + -2 = 0 ((15 + -12x) + 2x * (-5 + 4x)) + -2 = 0 (15 + -12x + (-5 * 2x + 4x * 2x)) + -2 = 0 (15 + -12x + (-10x + 8x2)) + -2 = 0 Combine like terms: -12x + -10x = -22x (15 + -22x + 8x2) + -2 = 0 Reorder the terms: 15 + -2 + -22x + 8x2 = 0 Combine like terms: 15 + -2 = 13 13 + -22x + 8x2 = 0 Solving 13 + -22x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.625 + -2.75x + x2 = 0 Move the constant term to the right: Add '-1.625' to each side of the equation. 1.625 + -2.75x + -1.625 + x2 = 0 + -1.625 Reorder the terms: 1.625 + -1.625 + -2.75x + x2 = 0 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + -2.75x + x2 = 0 + -1.625 -2.75x + x2 = 0 + -1.625 Combine like terms: 0 + -1.625 = -1.625 -2.75x + x2 = -1.625 The x term is -2.75x. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75x + 1.890625 + x2 = -1.625 + 1.890625 Reorder the terms: 1.890625 + -2.75x + x2 = -1.625 + 1.890625 Combine like terms: -1.625 + 1.890625 = 0.265625 1.890625 + -2.75x + x2 = 0.265625 Factor a perfect square on the left side: (x + -1.375)(x + -1.375) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (x + -1.375) equal to 0.515388203 and -0.515388203.Subproblem 1
x + -1.375 = 0.515388203 Simplifying x + -1.375 = 0.515388203 Reorder the terms: -1.375 + x = 0.515388203 Solving -1.375 + x = 0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = 0.515388203 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = 0.515388203 + 1.375 x = 0.515388203 + 1.375 Combine like terms: 0.515388203 + 1.375 = 1.890388203 x = 1.890388203 Simplifying x = 1.890388203Subproblem 2
x + -1.375 = -0.515388203 Simplifying x + -1.375 = -0.515388203 Reorder the terms: -1.375 + x = -0.515388203 Solving -1.375 + x = -0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = -0.515388203 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = -0.515388203 + 1.375 x = -0.515388203 + 1.375 Combine like terms: -0.515388203 + 1.375 = 0.859611797 x = 0.859611797 Simplifying x = 0.859611797Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.890388203, 0.859611797}
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